Aliases: C62:11Dic3, (C3xC62):2C4, C33:7(C22:C4), C3:2(C62:C4), C22:2(C33:C4), C32:5(C6.D4), (C6xC3:S3):6C4, (C2xC6):2(C32:C4), (C2xC3:S3):5Dic3, (C2xC3:S3).41D6, (C3xC3:S3).17D4, C6.14(C2xC32:C4), (C2xC33:C4):4C2, C3:S3.8(C3:D4), (C22xC3:S3).6S3, C2.7(C2xC33:C4), (C6xC3:S3).43C22, (C32xC6).21(C2xC4), (C3xC6).28(C2xDic3), (C2xC6xC3:S3).6C2, SmallGroup(432,641)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62:11Dic3
G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 904 in 152 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C23, C32, C32, Dic3, D6, C2xC6, C2xC6, C22:C4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C2xDic3, C22xS3, C22xC6, C33, C32:C4, S3xC6, C2xC3:S3, C2xC3:S3, C62, C62, C6.D4, C3xC3:S3, C3xC3:S3, C32xC6, C32xC6, C2xC32:C4, S3xC2xC6, C22xC3:S3, C33:C4, C6xC3:S3, C6xC3:S3, C3xC62, C62:C4, C2xC33:C4, C2xC6xC3:S3, C62:11Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, C2xDic3, C3:D4, C32:C4, C6.D4, C2xC32:C4, C33:C4, C62:C4, C2xC33:C4, C62:11Dic3
►On 24 points - transitive group 24T1286
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 6 2 4 3 5)(7 12 8 10 9 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 9 2 8 3 7)(4 12 5 11 6 10)(13 19 15 23 17 21)(14 24 16 22 18 20)
(1 23 8 13)(2 19 7 17)(3 21 9 15)(4 20 11 16)(5 22 10 14)(6 24 12 18)
G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,9,2,8,3,7)(4,12,5,11,6,10)(13,19,15,23,17,21)(14,24,16,22,18,20), (1,23,8,13)(2,19,7,17)(3,21,9,15)(4,20,11,16)(5,22,10,14)(6,24,12,18)>;
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), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,9,2,8,3,7)(4,12,5,11,6,10)(13,19,15,23,17,21)(14,24,16,22,18,20), (1,23,8,13)(2,19,7,17)(3,21,9,15)(4,20,11,16)(5,22,10,14)(6,24,12,18) );
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)], [(1,6,2,4,3,5),(7,12,8,10,9,11),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,9,2,8,3,7),(4,12,5,11,6,10),(13,19,15,23,17,21),(14,24,16,22,18,20)], [(1,23,8,13),(2,19,7,17),(3,21,9,15),(4,20,11,16),(5,22,10,14),(6,24,12,18)]])
G:=TransitiveGroup(24,1286);
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | ··· | 3G | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | ··· | 6U | 6V | 6W | 6X | 6Y |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 2 | 4 | ··· | 4 | 54 | 54 | 54 | 54 | 2 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Dic3 | D6 | Dic3 | C3:D4 | C32:C4 | C2xC32:C4 | C33:C4 | C62:C4 | C2xC33:C4 | C62:11Dic3 |
| kernel | C62:11Dic3 | C2xC33:C4 | C2xC6xC3:S3 | C6xC3:S3 | C3xC62 | C22xC3:S3 | C3xC3:S3 | C2xC3:S3 | C2xC3:S3 | C62 | C3:S3 | C2xC6 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C62:11Dic3 ►in GL4(F7) generated by
| 4 | 2 | 4 | 5 |
| 4 | 6 | 5 | 1 |
| 3 | 3 | 0 | 1 |
| 0 | 0 | 0 | 4 |
| 1 | 5 | 6 | 6 |
| 5 | 1 | 1 | 6 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 5 |
| 2 | 4 | 0 | 6 |
| 3 | 5 | 0 | 1 |
| 5 | 5 | 0 | 4 |
| 6 | 1 | 4 | 0 |
| 5 | 6 | 6 | 5 |
| 2 | 0 | 1 | 5 |
| 4 | 3 | 5 | 4 |
| 5 | 5 | 1 | 4 |
G:=sub<GL(4,GF(7))| [4,4,3,0,2,6,3,0,4,5,0,0,5,1,1,4],[1,5,0,0,5,1,0,0,6,1,6,0,6,6,0,5],[2,3,5,6,4,5,5,1,0,0,0,4,6,1,4,0],[5,2,4,5,6,0,3,5,6,1,5,1,5,5,4,4] >;
C62:11Dic3 in GAP, Magma, Sage, TeX
C_6^2\rtimes_{11}{\rm Dic}_3 % in TeX
G:=Group("C6^2:11Dic3"); // GroupNames label
G:=SmallGroup(432,641);
// by ID
G=gap.SmallGroup(432,641);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,2804,298,2693,1027,14118]);
// 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^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations