non-abelian, soluble, monomial
Aliases: C62:6Dic3, (C3xC6).9S4, (C6xA4).1S3, C32:A4:2C4, C6.12(C3:S4), (C3xA4):1Dic3, (C2xC62).9S3, C32:2(A4:C4), C22:(He3:3C4), C3.3(C6.7S4), C23.(He3:C2), C2.1(C32:S4), (C2xC32:A4).2C2, (C22xC6).1(C3:S3), (C2xC6).1(C3:Dic3), SmallGroup(432,260)
Series: Derived ►Chief ►Lower central ►Upper central
| C32:A4 — C62:6Dic3 |
Generators and relations for C62:6Dic3
G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, cac-1=ab-1, dad-1=a2b3, cbc-1=dbd-1=a3b4, dcd-1=c-1 >
Subgroups: 489 in 98 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2xC4, C23, C32, C32, Dic3, C12, A4, C2xC6, C2xC6, C22:C4, C3xC6, C3xC6, C2xDic3, C2xC12, C2xA4, C22xC6, C22xC6, He3, C3xDic3, C3xA4, C62, C62, C6.D4, C3xC22:C4, A4:C4, C2xHe3, C6xDic3, C6xA4, C2xC62, He3:3C4, C32:A4, C3xC6.D4, C3xA4:C4, C2xC32:A4, C62:6Dic3
Quotients: C1, C2, C4, S3, Dic3, C3:S3, S4, C3:Dic3, A4:C4, He3:C2, C3:S4, He3:3C4, C6.7S4, C32:S4, C62:6Dic3
►On 36 points
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)
(1 4 7 2 3 8)(5 11 10 6 12 9)(13 23 14 24 15 22)(16 21 17 19 18 20)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 17 30 2 20 27)(3 16 28 4 19 25)(5 13 35 6 24 32)(7 18 26 8 21 29)(9 22 34 10 14 31)(11 23 36 12 15 33)
(1 6 2 5)(3 11 4 12)(7 9 8 10)(13 27 24 30)(14 29 22 26)(15 25 23 28)(16 33 19 36)(17 35 20 32)(18 31 21 34)
G:=sub<Sym(36)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,4,7,2,3,8)(5,11,10,6,12,9)(13,23,14,24,15,22)(16,21,17,19,18,20)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,17,30,2,20,27)(3,16,28,4,19,25)(5,13,35,6,24,32)(7,18,26,8,21,29)(9,22,34,10,14,31)(11,23,36,12,15,33), (1,6,2,5)(3,11,4,12)(7,9,8,10)(13,27,24,30)(14,29,22,26)(15,25,23,28)(16,33,19,36)(17,35,20,32)(18,31,21,34)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,4,7,2,3,8)(5,11,10,6,12,9)(13,23,14,24,15,22)(16,21,17,19,18,20)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,17,30,2,20,27)(3,16,28,4,19,25)(5,13,35,6,24,32)(7,18,26,8,21,29)(9,22,34,10,14,31)(11,23,36,12,15,33), (1,6,2,5)(3,11,4,12)(7,9,8,10)(13,27,24,30)(14,29,22,26)(15,25,23,28)(16,33,19,36)(17,35,20,32)(18,31,21,34) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,4,7,2,3,8),(5,11,10,6,12,9),(13,23,14,24,15,22),(16,21,17,19,18,20),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(1,17,30,2,20,27),(3,16,28,4,19,25),(5,13,35,6,24,32),(7,18,26,8,21,29),(9,22,34,10,14,31),(11,23,36,12,15,33)], [(1,6,2,5),(3,11,4,12),(7,9,8,10),(13,27,24,30),(14,29,22,26),(15,25,23,28),(16,33,19,36),(17,35,20,32),(18,31,21,34)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6M | 6N | 6O | 6P | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 6 | 24 | 24 | 24 | 18 | 18 | 18 | 18 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 24 | 24 | 24 | 18 | ··· | 18 |
38 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | - | + | + | - | ||||||||
| image | C1 | C2 | C4 | S3 | S3 | Dic3 | Dic3 | S4 | A4:C4 | He3:C2 | He3:3C4 | C32:S4 | C62:6Dic3 | C3:S4 | C6.7S4 | C32:S4 | C62:6Dic3 |
| kernel | C62:6Dic3 | C2xC32:A4 | C32:A4 | C6xA4 | C2xC62 | C3xA4 | C62 | C3xC6 | C32 | C23 | C22 | C2 | C1 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 3 | 1 | 3 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C62:6Dic3 ►in GL3(F13) generated by
| 12 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 9 |
| 10 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 10 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 12 | 0 | 0 |
| 8 | 0 | 0 |
| 0 | 0 | 8 |
| 0 | 8 | 0 |
G:=sub<GL(3,GF(13))| [12,0,0,0,10,0,0,0,9],[10,0,0,0,3,0,0,0,10],[0,0,12,12,0,0,0,12,0],[8,0,0,0,0,8,0,8,0] >;
C62:6Dic3 in GAP, Magma, Sage, TeX
C_6^2\rtimes_6{\rm Dic}_3 % in TeX
G:=Group("C6^2:6Dic3"); // GroupNames label
G:=SmallGroup(432,260);
// by ID
G=gap.SmallGroup(432,260);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,14,170,675,353,9077,2287,5298,3989]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^2*b^3,c*b*c^-1=d*b*d^-1=a^3*b^4,d*c*d^-1=c^-1>;
// generators/relations