Automorphism Classes of Quadilateral Actions on Sufaces of Genus 6 

 
bdata:  6, 60, [ 2, 2, 2, 3 ] 
G = small group 5 of 13 group(s) of order 60 
Permutation group G acting on a set of cardinality 5
Order = 60 = 2^2 * 3 * 5
(1, 2, 3, 4, 5)
(1, 2, 3) 
9  generating vector(s)
<<(1, 2)(3, 4), (2, 4)(3, 5), (1, 4)(2, 3), (2, 5, 4)>, <false, false, false>> 
<<(1, 2)(3, 4), (2, 4)(3, 5), (1, 4)(3, 5), (2, 3, 4)>, <false, true, false>> 
<<(1, 2)(3, 4), (2, 4)(3, 5), (1, 2)(4, 5), (1, 3, 5)>, <false, true, true>> 
<<(1, 2)(3, 4), (2, 4)(3, 5), (2, 3)(4, 5), (1, 2, 5)>, <true, true, false>> 
<<(1, 2)(3, 4), (2, 4)(3, 5), (1, 2)(3, 5), (1, 3, 4)>, <false, false, false>> 
<<(1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(3, 5), (2, 3, 5)>, <true, false, true>> 
<<(1, 2)(3, 4), (1, 5)(3, 4), (1, 4)(2, 5), (1, 4, 5)>, <false, false, true>> 
<<(1, 2)(3, 4), (1, 5)(3, 4), (1, 2)(4, 5), (2, 5, 4)>, <false, false, false>> 
<<(1, 2)(3, 4), (1, 5)(3, 4), (1, 5)(2, 4), (1, 2, 4)>, <true, false, false>> 

 
bdata:  6, 28, [ 2, 2, 2, 7 ] 
G = small group 3 of 4 group(s) of order 28 
GrpPC : G of order 28 = 2^2 * 7
PC-Relations:
G.1^2 = Id(G), 
G.2^2 = Id(G), 
G.3^7 = Id(G), 
G.3^G.1 = G.3^6 
3  generating vector(s)
<<G.2, G.1 * G.2 * G.3^5, G.1, G.3^5>, <true, true, true>> 
<<G.1, G.2, G.1 * G.2 * G.3^6, G.3>, <true, true, true>> 
<<G.1, G.1 * G.2 * G.3^5, G.2, G.3^2>, <true, true, true>> 

 
bdata:  6, 24, [ 2, 2, 2, 12 ] 
G = small group 6 of 15 group(s) of order 24 
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
G.1^2 = Id(G), 
G.2^2 = G.3, 
G.3^2 = Id(G), 
G.4^3 = Id(G), 
G.2^G.1 = G.2 * G.3, 
G.4^G.1 = G.4^2 
3  generating vector(s)
<<G.3, G.1 * G.3 * G.4^2, G.1 * G.2 * G.3 * G.4, G.2 * G.4>, <true, true, true>> 
<<G.1, G.3, G.1 * G.2 * G.3 * G.4, G.2 * G.3 * G.4^2>, <true, true, true>> 
<<G.1, G.1 * G.2 * G.3 * G.4, G.3, G.2 * G.3 * G.4^2>, <true, true, true>> 

 
bdata:  6, 24, [ 2, 2, 3, 4 ] 
G = small group 6 of 15 group(s) of order 24 
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
G.1^2 = Id(G), 
G.2^2 = G.3, 
G.3^2 = Id(G), 
G.4^3 = Id(G), 
G.2^G.1 = G.2 * G.3, 
G.4^G.1 = G.4^2 
1  generating vector(s)
<<G.1, G.1 * G.2 * G.3 * G.4, G.4^2, G.2>, <true, true, true>> 

 
bdata:  6, 24, [ 2, 2, 3, 4 ] 
G = small group 8 of 15 group(s) of order 24 
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
G.1^2 = Id(G), 
G.2^2 = Id(G), 
G.3^2 = Id(G), 
G.4^3 = Id(G), 
G.2^G.1 = G.2 * G.3, 
G.4^G.1 = G.4^2 
2  generating vector(s)
<<G.2, G.1 * G.3 * G.4, G.4^2, G.1 * G.2 * G.3>, <true, true, true>> 
<<G.1, G.2 * G.3, G.4^2, G.1 * G.2 * G.4^2>, <true, true, true>> 

 
bdata:  6, 24, [ 2, 2, 3, 4 ] 
G = small group 12 of 15 group(s) of order 24 
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
G.1^2 = Id(G), 
G.2^3 = Id(G), 
G.3^2 = Id(G), 
G.4^2 = Id(G), 
G.2^G.1 = G.2^2, 
G.3^G.1 = G.4, 
G.3^G.2 = G.4, 
G.4^G.1 = G.3, 
G.4^G.2 = G.3 * G.4 
6  generating vector(s)
<<G.3, G.1 * G.3 * G.4, G.2^2 * G.3, G.1 * G.2^2 * G.3 * G.4>, <true, true, false>> 
<<G.3, G.1 * G.3 * G.4, G.2 * G.3 * G.4, G.1 * G.2 * G.3 * G.4>, <true, false, false>> 
<<G.3, G.1 * G.2^2, G.2 * G.3 * G.4, G.1 * G.4>, <true, false, true>> 
<<G.1, G.3 * G.4, G.2^2 * G.3, G.1 * G.2^2 * G.4>, <true, false, true>> 
<<G.1, G.4, G.2 * G.4, G.1 * G.2 * G.3 * G.4>, <false, true, true>> 
<<G.1, G.4, G.2^2 * G.3 * G.4, G.1 * G.2^2 * G.3 * G.4>, <false, false, true>> 

 
bdata:  6, 18, [ 2, 2, 3, 9 ] 
G = small group 1 of 5 group(s) of order 18 
GrpPC : G of order 18 = 2 * 3^2
PC-Relations:
G.1^2 = Id(G), 
G.2^3 = G.3, 
G.3^3 = Id(G), 
G.2^G.1 = G.2^2 * G.3^2, 
G.3^G.1 = G.3^2 
2  generating vector(s)
<<G.1, G.1 * G.2 * G.3, G.3, G.2^2>, <true, false, true>> 
<<G.1, G.1 * G.2 * G.3, G.3^2, G.2^2 * G.3^2>, <true, false, true>> 

 
bdata:  6, 20, [ 2, 2, 4, 4 ] 
G = small group 3 of 5 group(s) of order 20 
GrpPC : G of order 20 = 2^2 * 5
PC-Relations:
G.1^2 = G.2, 
G.2^2 = Id(G), 
G.3^5 = Id(G), 
G.3^G.1 = G.3^2, 
G.3^G.2 = G.3^4 
12 generating vector(s)
<<G.2, G.2, G.1 * G.3^4, G.1 * G.2 * G.3^3>, <false, false, false>> 
<<G.2, G.2, G.1 * G.2 * G.3, G.1 * G.3^3>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1, G.1 * G.2 * G.3^4>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.3^4, G.1 * G.2 * G.3^2>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.3^3, G.1 * G.2>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.3^2, G.1 * G.2 * G.3^3>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.3, G.1 * G.2 * G.3>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.2, G.1 * G.3^4>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.2 * G.3, G.1 * G.3^2>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.2 * G.3^3, G.1 * G.3^3>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.2 * G.3^2, G.1>, <false, false, false>> 
<<G.2, G.2 * G.3, G.1 * G.2 * G.3^4, G.1 * G.3>, <false, false, false>> 

 
bdata:  6, 16, [ 2, 2, 4, 8 ] 
G = small group 7 of 14 group(s) of order 16 
GrpPC : G of order 16 = 2^4
PC-Relations:
G.3^2 = G.4, 
G.2^G.1 = G.2 * G.3, 
G.3^G.1 = G.3 * G.4, 
G.3^G.2 = G.3 * G.4 
2  generating vector(s)
<<G.2, G.1 * G.3 * G.4, G.3 * G.4, G.1 * G.2 * G.4>, <true, false, true>> 
<<G.2, G.1 * G.3 * G.4, G.3, G.1 * G.2>, <true, false, true>> 

 
bdata:  6, 16, [ 2, 2, 4, 8 ] 
G = small group 8 of 14 group(s) of order 16 
GrpPC : G of order 16 = 2^4
PC-Relations:
G.1^2 = G.4, 
G.3^2 = G.4, 
G.2^G.1 = G.2 * G.3, 
G.3^G.1 = G.3 * G.4, 
G.3^G.2 = G.3 * G.4 
2  generating vector(s)
<<G.4, G.2 * G.3 * G.4, G.1 * G.3 * G.4, G.1 * G.2>, <true, true, true>> 
<<G.2, G.4, G.1 * G.3 * G.4, G.1 * G.2 * G.3>, <true, true, true>> 

 
bdata:  6, 14, [ 2, 2, 7, 7 ] 
G = small group 1 of 2 group(s) of order 14 
GrpPC : G of order 14 = 2 * 7
PC-Relations:
G.1^2 = Id(G), 
G.2^7 = Id(G), 
G.2^G.1 = G.2^6 
6  generating vector(s)
<<G.1, G.1 * G.2, G.2, G.2^5>, <true, false, true>> 
<<G.1, G.1 * G.2, G.2^2, G.2^4>, <true, false, true>> 
<<G.1, G.1 * G.2, G.2^5, G.2>, <true, false, true>> 
<<G.1, G.1 * G.2, G.2^3, G.2^3>, <true, false, true>> 
<<G.1, G.1 * G.2, G.2^4, G.2^2>, <true, false, true>> 
<<G.1, G.1, G.2, G.2^6>, <true, false, true>> 

 
bdata:  6, 14, [ 2, 2, 7, 7 ] 
G = small group 2 of 2 group(s) of order 14 
GrpPC : G of order 14 = 2 * 7
PC-Relations:
G.1^2 = Id(G), 
G.2^7 = Id(G) 
1  generating vector(s)
<<G.1, G.1, G.2, G.2^6>, <true, true, true>> 

 
bdata:  6, 12, [ 2, 2, 12, 12 ] 
G = small group 2 of 5 group(s) of order 12 
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
G.1^2 = G.3, 
G.2^3 = Id(G), 
G.3^2 = Id(G) 
1  generating vector(s)
<<G.3, G.3, G.1 * G.2, G.1 * G.2^2 * G.3>, <true, true, true>> 

 
bdata:  6, 12, [ 2, 3, 4, 12 ] 
G = small group 2 of 5 group(s) of order 12 
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
G.1^2 = G.3, 
G.2^3 = Id(G), 
G.3^2 = Id(G) 
1  generating vector(s)
<<G.3, G.2, G.1, G.1 * G.2^2>, <true, true, true>> 

 
bdata:  6, 12, [ 2, 3, 6, 6 ] 
G = small group 5 of 5 group(s) of order 12 
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
G.1^2 = Id(G), 
G.2^2 = Id(G), 
G.3^3 = Id(G) 
1  generating vector(s)
<<G.2, G.3, G.1 * G.3, G.1 * G.2 * G.3>, <true, true, true>> 

 
bdata:  6, 12, [ 2, 4, 4, 6 ] 
G = small group 1 of 5 group(s) of order 12 
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
G.1^2 = G.2, 
G.2^2 = Id(G), 
G.3^3 = Id(G), 
G.3^G.1 = G.3^2 
1  generating vector(s)
<<G.2, G.1, G.1 * G.2 * G.3, G.2 * G.3^2>, <true, true, true>> 

 
bdata:  6, 10, [ 2, 5, 5, 10 ] 
G = small group 2 of 2 group(s) of order 10 
GrpPC : G of order 10 = 2 * 5
PC-Relations:
G.1^2 = Id(G), 
G.2^5 = Id(G) 
3  generating vector(s)
<<G.1, G.2, G.2, G.1 * G.2^3>, <true, true, true>> 
<<G.1, G.2, G.2^2, G.1 * G.2^2>, <true, true, true>> 
<<G.1, G.2, G.2^3, G.1 * G.2>, <true, true, true>> 

 
bdata:  6, 12, [ 3, 3, 4, 4 ] 
G = small group 1 of 5 group(s) of order 12 
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
G.1^2 = G.2, 
G.2^2 = Id(G), 
G.3^3 = Id(G), 
G.3^G.1 = G.3^2 
2  generating vector(s)
<<G.3, G.3, G.1, G.1 * G.2 * G.3>, <true, false, true>> 
<<G.3, G.3^2, G.1, G.1 * G.2>, <true, false, true>> 

 
bdata:  6, 12, [ 3, 3, 4, 4 ] 
G = small group 2 of 5 group(s) of order 12 
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
G.1^2 = G.3, 
G.2^3 = Id(G), 
G.3^2 = Id(G) 
1  generating vector(s)
<<G.2, G.2^2, G.1, G.1 * G.3>, <true, true, true>> 

 
bdata:  6,  9, [ 3, 3, 9, 9 ] 
G = small group 1 of 2 group(s) of order 9 
GrpPC : G of order 9 = 3^2
PC-Relations:
G.1^3 = G.2 
4  generating vector(s)
<<G.2, G.2, G.1, G.1^2>, <true, true, true>> 
<<G.2, G.2, G.1^2, G.1>, <true, true, true>> 
<<G.2, G.2^2, G.1, G.1^2 * G.2^2>, <true, true, true>> 
<<G.2, G.2^2, G.1^2, G.1 * G.2^2>, <true, true, true>> 

 
bdata:  6,  8, [ 4, 4, 8, 8 ] 
G = small group 1 of 5 group(s) of order 8 
GrpPC : G of order 8 = 2^3
PC-Relations:
G.1^2 = G.2, 
G.2^2 = G.3 
4  generating vector(s)
<<G.2, G.2, G.1, G.1 * G.2>, <true, true, true>> 
<<G.2, G.2, G.1 * G.2, G.1>, <true, true, true>> 
<<G.2, G.2 * G.3, G.1, G.1 * G.2 * G.3>, <true, true, true>> 
<<G.2, G.2 * G.3, G.1 * G.2, G.1 * G.3>, <true, true, true>> 

 
bdata:  6,  7, [ 7, 7, 7, 7 ] 
G = small group 1 of 1 group(s) of order 7 
GrpPC : G of order 7
PC-Relations:
G.1^7 = Id(G) 
31 generating vector(s)
<<G.1, G.1, G.1, G.1^4>, <true, true, true>> 
<<G.1, G.1, G.1^2, G.1^3>, <true, true, true>> 
<<G.1, G.1, G.1^3, G.1^2>, <true, true, true>> 
<<G.1, G.1, G.1^4, G.1>, <true, true, true>> 
<<G.1, G.1, G.1^6, G.1^6>, <true, true, true>> 
<<G.1, G.1^3, G.1, G.1^2>, <true, true, true>> 
<<G.1, G.1^3, G.1^2, G.1>, <true, true, true>> 
<<G.1, G.1^3, G.1^4, G.1^6>, <true, true, true>> 
<<G.1, G.1^3, G.1^5, G.1^5>, <true, true, true>> 
<<G.1, G.1^3, G.1^6, G.1^4>, <true, true, true>> 
<<G.1, G.1^2, G.1, G.1^3>, <true, true, true>> 
<<G.1, G.1^2, G.1^2, G.1^2>, <true, true, true>> 
<<G.1, G.1^2, G.1^3, G.1>, <true, true, true>> 
<<G.1, G.1^2, G.1^5, G.1^6>, <true, true, true>> 
<<G.1, G.1^2, G.1^6, G.1^5>, <true, true, true>> 
<<G.1, G.1^5, G.1^2, G.1^6>, <true, true, true>> 
<<G.1, G.1^5, G.1^3, G.1^5>, <true, true, true>> 
<<G.1, G.1^5, G.1^4, G.1^4>, <true, true, true>> 
<<G.1, G.1^5, G.1^5, G.1^3>, <true, true, true>> 
<<G.1, G.1^5, G.1^6, G.1^2>, <true, true, true>> 
<<G.1, G.1^4, G.1, G.1>, <true, true, true>> 
<<G.1, G.1^4, G.1^3, G.1^6>, <true, true, true>> 
<<G.1, G.1^4, G.1^4, G.1^5>, <true, true, true>> 
<<G.1, G.1^4, G.1^5, G.1^4>, <true, true, true>> 
<<G.1, G.1^4, G.1^6, G.1^3>, <true, true, true>> 
<<G.1, G.1^6, G.1, G.1^6>, <true, true, true>> 
<<G.1, G.1^6, G.1^2, G.1^5>, <true, true, true>> 
<<G.1, G.1^6, G.1^3, G.1^4>, <true, true, true>> 
<<G.1, G.1^6, G.1^4, G.1^3>, <true, true, true>> 
<<G.1, G.1^6, G.1^5, G.1^2>, <true, true, true>> 
<<G.1, G.1^6, G.1^6, G.1>, <true, true, true>>