metabelian, supersoluble, monomial
Aliases: C12.78D12, C6.4(S3×C8), C3⋊1(D6⋊C8), C6.1(D6⋊C4), C6.2(C8⋊S3), (C3×C12).108D4, (C2×C12).296D6, C32⋊5(C22⋊C8), C62.26(C2×C4), (C3×C6).6M4(2), C12.77(C3⋊D4), C4.27(C3⋊D12), (C6×C12).201C22, C2.4(C12.29D6), C2.2(C12.31D6), C2.1(C6.D12), C22.9(C6.D6), (C6×C3⋊C8)⋊2C2, (C2×C3⋊S3)⋊3C8, (C2×C3⋊C8)⋊10S3, (C2×C4).129S32, (C2×C6).27(C4×S3), (C3×C6).19(C2×C8), (C22×C3⋊S3).6C4, (C2×C3⋊Dic3).8C4, (C3×C6).24(C22⋊C4), (C2×C4×C3⋊S3).11C2, SmallGroup(288,205)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.78D12
G = < a,b,c | a12=1, b12=c2=a6, bab-1=cac-1=a5, cbc-1=a3b11 >
Subgroups: 530 in 127 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C22×C4, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊C8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C2×C3⋊C8, C2×C24, S3×C2×C4, C3×C3⋊C8, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊C8, C6×C3⋊C8, C2×C4×C3⋊S3, C12.78D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), C4×S3, D12, C3⋊D4, C22⋊C8, S32, S3×C8, C8⋊S3, D6⋊C4, C6.D6, C3⋊D12, D6⋊C8, C12.29D6, C12.31D6, C6.D12, C12.78D12
►On 48 points
(1 27 5 31 9 35 13 39 17 43 21 47)(2 36 22 32 18 28 14 48 10 44 6 40)(3 29 7 33 11 37 15 41 19 45 23 25)(4 38 24 34 20 30 16 26 12 46 8 42)
(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)
(1 3 13 15)(2 44 14 32)(4 30 16 42)(5 23 17 11)(6 40 18 28)(7 9 19 21)(8 26 20 38)(10 36 22 48)(12 46 24 34)(25 27 37 39)(29 47 41 35)(31 33 43 45)
G:=sub<Sym(48)| (1,27,5,31,9,35,13,39,17,43,21,47)(2,36,22,32,18,28,14,48,10,44,6,40)(3,29,7,33,11,37,15,41,19,45,23,25)(4,38,24,34,20,30,16,26,12,46,8,42), (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), (1,3,13,15)(2,44,14,32)(4,30,16,42)(5,23,17,11)(6,40,18,28)(7,9,19,21)(8,26,20,38)(10,36,22,48)(12,46,24,34)(25,27,37,39)(29,47,41,35)(31,33,43,45)>;
G:=Group( (1,27,5,31,9,35,13,39,17,43,21,47)(2,36,22,32,18,28,14,48,10,44,6,40)(3,29,7,33,11,37,15,41,19,45,23,25)(4,38,24,34,20,30,16,26,12,46,8,42), (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), (1,3,13,15)(2,44,14,32)(4,30,16,42)(5,23,17,11)(6,40,18,28)(7,9,19,21)(8,26,20,38)(10,36,22,48)(12,46,24,34)(25,27,37,39)(29,47,41,35)(31,33,43,45) );
G=PermutationGroup([[(1,27,5,31,9,35,13,39,17,43,21,47),(2,36,22,32,18,28,14,48,10,44,6,40),(3,29,7,33,11,37,15,41,19,45,23,25),(4,38,24,34,20,30,16,26,12,46,8,42)], [(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)], [(1,3,13,15),(2,44,14,32),(4,30,16,42),(5,23,17,11),(6,40,18,28),(7,9,19,21),(8,26,20,38),(10,36,22,48),(12,46,24,34),(25,27,37,39),(29,47,41,35),(31,33,43,45)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 8A | ··· | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | D12 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | S32 | C3⋊D12 | C6.D6 | C12.29D6 | C12.31D6 |
| kernel | C12.78D12 | C6×C3⋊C8 | C2×C4×C3⋊S3 | C2×C3⋊Dic3 | C22×C3⋊S3 | C2×C3⋊S3 | C2×C3⋊C8 | C3×C12 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C6 | C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C12.78D12 ►in GL6(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 27 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 71 | 0 | 0 |
| 0 | 0 | 72 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 63 | 10 |
| 0 | 0 | 0 | 0 | 63 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 46 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 |
G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[46,27,0,0,0,0,0,27,0,0,0,0,0,0,46,72,0,0,0,0,71,27,0,0,0,0,0,0,63,63,0,0,0,0,10,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,46,0,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;
C12.78D12 in GAP, Magma, Sage, TeX
C_{12}._{78}D_{12} % in TeX
G:=Group("C12.78D12"); // GroupNames label
G:=SmallGroup(288,205);
// by ID
G=gap.SmallGroup(288,205);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=c^2=a^6,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations