metabelian, supersoluble, monomial
Aliases: C12.19D12, (C2xC12).90D6, (C3xC12).51D4, C6.27(D6:C4), (C3xM4(2)):9S3, C62.40(C2xC4), C12.58D6:4C2, M4(2):3(C3:S3), (C6xC12).57C22, C4.11(C12:S3), C32:6(C4.D4), C3:2(C12.46D4), C12.116(C3:D4), C4.21(C32:7D4), C2.9(C6.11D12), (C32xM4(2)):13C2, (C2xC6).14(C4xS3), C22.4(C4xC3:S3), (C22xC3:S3).2C4, (C2xC12:S3).10C2, (C3xC6).58(C22:C4), (C2xC4).1(C2xC3:S3), SmallGroup(288,298)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.19D12
G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=a7, cac-1=a5, cbc-1=a3b11 >
Subgroups: 748 in 138 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2xC4, D4, C23, C32, C12, D6, C2xC6, M4(2), M4(2), C2xD4, C3:S3, C3xC6, C3xC6, C3:C8, C24, D12, C2xC12, C22xS3, C4.D4, C3xC12, C2xC3:S3, C62, C4.Dic3, C3xM4(2), C2xD12, C32:4C8, C3xC24, C12:S3, C6xC12, C22xC3:S3, C12.46D4, C12.58D6, C32xM4(2), C2xC12:S3, C12.19D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, C3:S3, C4xS3, D12, C3:D4, C4.D4, C2xC3:S3, D6:C4, C4xC3:S3, C12:S3, C32:7D4, C12.46D4, C6.11D12, C12.19D12
►On 72 points
(1 35 56 7 41 62 13 47 68 19 29 50)(2 48 57 20 42 51 14 36 69 8 30 63)(3 37 58 9 43 64 15 25 70 21 31 52)(4 26 59 22 44 53 16 38 71 10 32 65)(5 39 60 11 45 66 17 27 72 23 33 54)(6 28 61 24 46 55 18 40 49 12 34 67)
(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 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 12 19 18 13 24 7 6)(2 5 8 23 14 17 20 11)(3 10 21 16 15 22 9 4)(25 71 43 53 37 59 31 65)(26 64 32 58 38 52 44 70)(27 69 45 51 39 57 33 63)(28 62 34 56 40 50 46 68)(29 67 47 49 41 55 35 61)(30 60 36 54 42 72 48 66)
G:=sub<Sym(72)| (1,35,56,7,41,62,13,47,68,19,29,50)(2,48,57,20,42,51,14,36,69,8,30,63)(3,37,58,9,43,64,15,25,70,21,31,52)(4,26,59,22,44,53,16,38,71,10,32,65)(5,39,60,11,45,66,17,27,72,23,33,54)(6,28,61,24,46,55,18,40,49,12,34,67), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,12,19,18,13,24,7,6)(2,5,8,23,14,17,20,11)(3,10,21,16,15,22,9,4)(25,71,43,53,37,59,31,65)(26,64,32,58,38,52,44,70)(27,69,45,51,39,57,33,63)(28,62,34,56,40,50,46,68)(29,67,47,49,41,55,35,61)(30,60,36,54,42,72,48,66)>;
G:=Group( (1,35,56,7,41,62,13,47,68,19,29,50)(2,48,57,20,42,51,14,36,69,8,30,63)(3,37,58,9,43,64,15,25,70,21,31,52)(4,26,59,22,44,53,16,38,71,10,32,65)(5,39,60,11,45,66,17,27,72,23,33,54)(6,28,61,24,46,55,18,40,49,12,34,67), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,12,19,18,13,24,7,6)(2,5,8,23,14,17,20,11)(3,10,21,16,15,22,9,4)(25,71,43,53,37,59,31,65)(26,64,32,58,38,52,44,70)(27,69,45,51,39,57,33,63)(28,62,34,56,40,50,46,68)(29,67,47,49,41,55,35,61)(30,60,36,54,42,72,48,66) );
G=PermutationGroup([[(1,35,56,7,41,62,13,47,68,19,29,50),(2,48,57,20,42,51,14,36,69,8,30,63),(3,37,58,9,43,64,15,25,70,21,31,52),(4,26,59,22,44,53,16,38,71,10,32,65),(5,39,60,11,45,66,17,27,72,23,33,54),(6,28,61,24,46,55,18,40,49,12,34,67)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,12,19,18,13,24,7,6),(2,5,8,23,14,17,20,11),(3,10,21,16,15,22,9,4),(25,71,43,53,37,59,31,65),(26,64,32,58,38,52,44,70),(27,69,45,51,39,57,33,63),(28,62,34,56,40,50,46,68),(29,67,47,49,41,55,35,61),(30,60,36,54,42,72,48,66)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D12 | C3:D4 | C4xS3 | C4.D4 | C12.46D4 |
| kernel | C12.19D12 | C12.58D6 | C32xM4(2) | C2xC12:S3 | C22xC3:S3 | C3xM4(2) | C3xC12 | C2xC12 | C12 | C12 | C2xC6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 8 | 8 | 1 | 8 |
Matrix representation of C12.19D12 ►in GL6(F73)
| 36 | 47 | 0 | 0 | 0 | 0 |
| 26 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 56 | 25 | 59 | 7 |
| 0 | 0 | 22 | 40 | 66 | 66 |
| 26 | 37 | 0 | 0 | 0 | 0 |
| 36 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 4 | 72 | 2 |
| 0 | 0 | 22 | 41 | 71 | 1 |
| 0 | 0 | 18 | 44 | 41 | 59 |
| 0 | 0 | 65 | 57 | 40 | 33 |
| 37 | 47 | 0 | 0 | 0 | 0 |
| 47 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 52 | 0 |
| 0 | 0 | 6 | 23 | 52 | 21 |
| 0 | 0 | 64 | 19 | 20 | 35 |
| 0 | 0 | 69 | 5 | 49 | 12 |
G:=sub<GL(6,GF(73))| [36,26,0,0,0,0,47,36,0,0,0,0,0,0,14,7,56,22,0,0,66,7,25,40,0,0,0,0,59,66,0,0,0,0,7,66],[26,36,0,0,0,0,37,26,0,0,0,0,0,0,31,22,18,65,0,0,4,41,44,57,0,0,72,71,41,40,0,0,2,1,59,33],[37,47,0,0,0,0,47,36,0,0,0,0,0,0,18,6,64,69,0,0,35,23,19,5,0,0,52,52,20,49,0,0,0,21,35,12] >;
C12.19D12 in GAP, Magma, Sage, TeX
C_{12}._{19}D_{12} % in TeX
G:=Group("C12.19D12"); // GroupNames label
G:=SmallGroup(288,298);
// by ID
G=gap.SmallGroup(288,298);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,100,346,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations