metabelian, supersoluble, monomial
Aliases: Dic6.24D6, C62.1C23, C32⋊32- 1+4, D4.11S32, C3⋊D4.1D6, D4⋊2S3⋊4S3, C3⋊3(Q8○D12), (S3×Dic6)⋊9C2, (C4×S3).15D6, (C3×D4).23D6, D6.4D6⋊4C2, D6.D6⋊5C2, D6.3D6⋊2C2, (S3×C6).8C23, (C3×C6).16C24, C6.16(S3×C23), D6.9(C22×S3), C12.D6⋊6C2, C32⋊7D4.C22, (C3×C12).29C23, C12.29(C22×S3), (C2×Dic3).50D6, Dic3.D6⋊11C2, (S3×C12).32C22, C3⋊D12.2C22, D6⋊S3.8C22, C3⋊Dic3.19C23, (S3×Dic3).3C22, C6.D6.6C22, C32⋊2Q8.9C22, Dic3.21(C22×S3), (C3×Dic6).29C22, (C6×Dic3).48C22, (C3×Dic3).11C23, (D4×C32).21C22, C32⋊4Q8.21C22, C4.29(C2×S32), C22.1(C2×S32), C2.18(C22×S32), (C3×C3⋊D4).C22, (C3×D4⋊2S3)⋊7C2, (C2×C6).2(C22×S3), (C2×C32⋊2Q8)⋊16C2, (C2×C3⋊S3).22C23, (C4×C3⋊S3).42C22, (C2×C3⋊Dic3).104C22, SmallGroup(288,957)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.24D6
G = < a,b,c,d | a12=c6=1, b2=d2=a6, bab-1=a-1, cac-1=dad-1=a7, bc=cb, bd=db, dcd-1=a6c-1 >
Subgroups: 1018 in 312 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, 2- 1+4, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C2×Dic6, C4○D12, D4⋊2S3, D4⋊2S3, S3×Q8, C3×C4○D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C32⋊2Q8, C32⋊2Q8, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, Q8○D12, S3×Dic6, Dic3.D6, D6.D6, D6.3D6, D6.4D6, C2×C32⋊2Q8, C3×D4⋊2S3, C12.D6, Dic6.24D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S32, S3×C23, C2×S32, Q8○D12, C22×S32, Dic6.24D6
►On 48 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)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 28 7 34)(2 27 8 33)(3 26 9 32)(4 25 10 31)(5 36 11 30)(6 35 12 29)(13 46 19 40)(14 45 20 39)(15 44 21 38)(16 43 22 37)(17 42 23 48)(18 41 24 47)
(1 18 9 14 5 22)(2 13 10 21 6 17)(3 20 11 16 7 24)(4 15 12 23 8 19)(25 44 29 48 33 40)(26 39 30 43 34 47)(27 46 31 38 35 42)(28 41 32 45 36 37)
(1 26 7 32)(2 33 8 27)(3 28 9 34)(4 35 10 29)(5 30 11 36)(6 25 12 31)(13 42 19 48)(14 37 20 43)(15 44 21 38)(16 39 22 45)(17 46 23 40)(18 41 24 47)
G:=sub<Sym(48)| (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), (1,28,7,34)(2,27,8,33)(3,26,9,32)(4,25,10,31)(5,36,11,30)(6,35,12,29)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47), (1,18,9,14,5,22)(2,13,10,21,6,17)(3,20,11,16,7,24)(4,15,12,23,8,19)(25,44,29,48,33,40)(26,39,30,43,34,47)(27,46,31,38,35,42)(28,41,32,45,36,37), (1,26,7,32)(2,33,8,27)(3,28,9,34)(4,35,10,29)(5,30,11,36)(6,25,12,31)(13,42,19,48)(14,37,20,43)(15,44,21,38)(16,39,22,45)(17,46,23,40)(18,41,24,47)>;
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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,28,7,34)(2,27,8,33)(3,26,9,32)(4,25,10,31)(5,36,11,30)(6,35,12,29)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47), (1,18,9,14,5,22)(2,13,10,21,6,17)(3,20,11,16,7,24)(4,15,12,23,8,19)(25,44,29,48,33,40)(26,39,30,43,34,47)(27,46,31,38,35,42)(28,41,32,45,36,37), (1,26,7,32)(2,33,8,27)(3,28,9,34)(4,35,10,29)(5,30,11,36)(6,25,12,31)(13,42,19,48)(14,37,20,43)(15,44,21,38)(16,39,22,45)(17,46,23,40)(18,41,24,47) );
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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,28,7,34),(2,27,8,33),(3,26,9,32),(4,25,10,31),(5,36,11,30),(6,35,12,29),(13,46,19,40),(14,45,20,39),(15,44,21,38),(16,43,22,37),(17,42,23,48),(18,41,24,47)], [(1,18,9,14,5,22),(2,13,10,21,6,17),(3,20,11,16,7,24),(4,15,12,23,8,19),(25,44,29,48,33,40),(26,39,30,43,34,47),(27,46,31,38,35,42),(28,41,32,45,36,37)], [(1,26,7,32),(2,33,8,27),(3,28,9,34),(4,35,10,29),(5,30,11,36),(6,25,12,31),(13,42,19,48),(14,37,20,43),(15,44,21,38),(16,39,22,45),(17,46,23,40),(18,41,24,47)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 4A | 4B | ··· | 4G | 4H | 4I | 4J | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 2 | 2 | 4 | 2 | 6 | ··· | 6 | 18 | 18 | 18 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 12 | 12 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 12 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | 2- 1+4 | S32 | C2×S32 | C2×S32 | Q8○D12 | Dic6.24D6 |
| kernel | Dic6.24D6 | S3×Dic6 | Dic3.D6 | D6.D6 | D6.3D6 | D6.4D6 | C2×C32⋊2Q8 | C3×D4⋊2S3 | C12.D6 | D4⋊2S3 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | C32 | D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 1 | 1 | 1 | 2 | 4 | 1 |
Matrix representation of Dic6.24D6 ►in GL6(𝔽13)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 7 | 0 |
| 0 | 0 | 11 | 1 | 0 | 7 |
| 0 | 0 | 4 | 0 | 1 | 11 |
| 0 | 0 | 0 | 4 | 2 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 7 | 12 | 8 | 0 |
| 0 | 0 | 1 | 6 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 11 | 2 | 11 | 11 |
| 0 | 0 | 11 | 0 | 2 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,11,4,0,0,0,2,1,0,4,0,0,7,0,1,2,0,0,0,7,11,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,5,0,7,1,0,0,0,5,12,6,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,11,11,11,0,0,2,4,2,0,0,0,0,0,11,2,0,0,0,0,11,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,0,5,5,0,0,0,0,0,8] >;
Dic6.24D6 in GAP, Magma, Sage, TeX
{\rm Dic}_6._{24}D_6 % in TeX
G:=Group("Dic6.24D6"); // GroupNames label
G:=SmallGroup(288,957);
// by ID
G=gap.SmallGroup(288,957);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,185,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^6=1,b^2=d^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*c^-1>;
// generators/relations