metabelian, supersoluble, monomial
Aliases: C12.25D12, C62.52D4, Dic6.29D6, C3⋊C8.4D6, (C2×Dic6)⋊9S3, (C6×Dic6)⋊4C2, (C2×C6).62D12, C6.72(C2×D12), (C3×C12).70D4, (C2×C12).122D6, C3⋊4(C8.D6), C4.Dic3⋊10S3, C32⋊3Q16⋊6C2, C32⋊5SD16⋊4C2, C12.49(C3⋊D4), (C6×C12).82C22, (C3×C12).69C23, C12.85(C22×S3), C32⋊8(C8.C22), C4.24(C3⋊D12), C3⋊1(Q8.11D6), C12.59D6.5C2, C12⋊S3.28C22, C22.5(C3⋊D12), (C3×Dic6).36C22, C32⋊4Q8.28C22, C4.57(C2×S32), (C2×C4).10S32, C6.8(C2×C3⋊D4), (C3×C6).73(C2×D4), (C3×C3⋊C8).4C22, (C3×C4.Dic3)⋊4C2, C2.12(C2×C3⋊D12), (C2×C6).21(C3⋊D4), SmallGroup(288,481)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.29D6
G = < a,b,c,d | a12=c6=1, b2=a6, d2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >
Subgroups: 570 in 140 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C24⋊C2, Dic12, C4.Dic3, Q8⋊2S3, C3⋊Q16, C3×M4(2), C2×Dic6, C4○D12, C6×Q8, C3×C3⋊C8, C3×Dic6, C3×Dic6, C6×Dic3, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C8.D6, Q8.11D6, C32⋊5SD16, C32⋊3Q16, C3×C4.Dic3, C6×Dic6, C12.59D6, Dic6.29D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, C8.C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C8.D6, Q8.11D6, C2×C3⋊D12, Dic6.29D6
►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 33 7 27)(2 32 8 26)(3 31 9 25)(4 30 10 36)(5 29 11 35)(6 28 12 34)(13 42 19 48)(14 41 20 47)(15 40 21 46)(16 39 22 45)(17 38 23 44)(18 37 24 43)
(1 15 9 23 5 19)(2 16 10 24 6 20)(3 17 11 13 7 21)(4 18 12 14 8 22)(25 44 29 48 33 40)(26 45 30 37 34 41)(27 46 31 38 35 42)(28 47 32 39 36 43)
(1 45 4 48 7 39 10 42)(2 46 5 37 8 40 11 43)(3 47 6 38 9 41 12 44)(13 30 16 33 19 36 22 27)(14 31 17 34 20 25 23 28)(15 32 18 35 21 26 24 29)
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), (1,33,7,27)(2,32,8,26)(3,31,9,25)(4,30,10,36)(5,29,11,35)(6,28,12,34)(13,42,19,48)(14,41,20,47)(15,40,21,46)(16,39,22,45)(17,38,23,44)(18,37,24,43), (1,15,9,23,5,19)(2,16,10,24,6,20)(3,17,11,13,7,21)(4,18,12,14,8,22)(25,44,29,48,33,40)(26,45,30,37,34,41)(27,46,31,38,35,42)(28,47,32,39,36,43), (1,45,4,48,7,39,10,42)(2,46,5,37,8,40,11,43)(3,47,6,38,9,41,12,44)(13,30,16,33,19,36,22,27)(14,31,17,34,20,25,23,28)(15,32,18,35,21,26,24,29)>;
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), (1,33,7,27)(2,32,8,26)(3,31,9,25)(4,30,10,36)(5,29,11,35)(6,28,12,34)(13,42,19,48)(14,41,20,47)(15,40,21,46)(16,39,22,45)(17,38,23,44)(18,37,24,43), (1,15,9,23,5,19)(2,16,10,24,6,20)(3,17,11,13,7,21)(4,18,12,14,8,22)(25,44,29,48,33,40)(26,45,30,37,34,41)(27,46,31,38,35,42)(28,47,32,39,36,43), (1,45,4,48,7,39,10,42)(2,46,5,37,8,40,11,43)(3,47,6,38,9,41,12,44)(13,30,16,33,19,36,22,27)(14,31,17,34,20,25,23,28)(15,32,18,35,21,26,24,29) );
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)], [(1,33,7,27),(2,32,8,26),(3,31,9,25),(4,30,10,36),(5,29,11,35),(6,28,12,34),(13,42,19,48),(14,41,20,47),(15,40,21,46),(16,39,22,45),(17,38,23,44),(18,37,24,43)], [(1,15,9,23,5,19),(2,16,10,24,6,20),(3,17,11,13,7,21),(4,18,12,14,8,22),(25,44,29,48,33,40),(26,45,30,37,34,41),(27,46,31,38,35,42),(28,47,32,39,36,43)], [(1,45,4,48,7,39,10,42),(2,46,5,37,8,40,11,43),(3,47,6,38,9,41,12,44),(13,30,16,33,19,36,22,27),(14,31,17,34,20,25,23,28),(15,32,18,35,21,26,24,29)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 12A | 12B | 12C | ··· | 12I | 12J | 12K | 12L | 12M | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 36 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | C3⋊D4 | D12 | C3⋊D4 | C8.C22 | S32 | C3⋊D12 | C2×S32 | C3⋊D12 | C8.D6 | Q8.11D6 | Dic6.29D6 |
| kernel | Dic6.29D6 | C32⋊5SD16 | C32⋊3Q16 | C3×C4.Dic3 | C6×Dic6 | C12.59D6 | C4.Dic3 | C2×Dic6 | C3×C12 | C62 | C3⋊C8 | Dic6 | C2×C12 | C12 | C12 | C2×C6 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of Dic6.29D6 ►in GL4(𝔽73) generated by
| 14 | 7 | 0 | 0 |
| 66 | 7 | 0 | 0 |
| 0 | 0 | 14 | 7 |
| 0 | 0 | 66 | 7 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 30 | 43 | 0 | 0 |
| 0 | 0 | 43 | 30 |
| 0 | 0 | 43 | 13 |
| 0 | 0 | 30 | 43 |
| 0 | 0 | 30 | 60 |
| 0 | 27 | 0 | 0 |
| 46 | 46 | 0 | 0 |
G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,14,66,0,0,7,7],[0,0,1,72,0,0,0,72,72,1,0,0,0,1,0,0],[13,30,0,0,43,43,0,0,0,0,43,43,0,0,30,13],[0,0,0,46,0,0,27,46,30,30,0,0,43,60,0,0] >;
Dic6.29D6 in GAP, Magma, Sage, TeX
{\rm Dic}_6._{29}D_6 % in TeX
G:=Group("Dic6.29D6"); // GroupNames label
G:=SmallGroup(288,481);
// by ID
G=gap.SmallGroup(288,481);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,64,219,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^6=1,b^2=a^6,d^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;
// generators/relations