metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.40D4, Dic6.40D4, M4(2).17D6, C4○D4.24D6, (C3×D4).17D4, C4.107(S3×D4), C8.C22⋊3S3, Q8○D12.2C2, (C3×Q8).17D4, (C2×Q8).95D6, D12⋊C4⋊8C2, C6.66C22≀C2, C12.201(C2×D4), (C2×Dic3).6D4, C22.38(S3×D4), Q8⋊3Dic3⋊9C2, Dic3⋊Q8⋊7C2, Q8.11D6⋊6C2, C12.47D4⋊8C2, D4.12(C3⋊D4), C3⋊4(D4.10D4), (C2×C12).20C23, Q8.19(C3⋊D4), (C6×Q8).98C22, C2.34(C23⋊2D6), C4○D12.26C22, (C4×Dic3).59C22, C4.Dic3.29C22, (C2×Dic6).137C22, (C3×M4(2)).14C22, (C2×C6).37(C2×D4), C4.57(C2×C3⋊D4), (C3×C8.C22)⋊7C2, (C2×C4).20(C22×S3), (C3×C4○D4).18C22, SmallGroup(192,764)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.40D4
G = < a,b,c,d | a12=b2=1, c4=d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=a6c3 >
Subgroups: 400 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, C8.C22, 2- 1+4, C4.Dic3, C4×Dic3, Dic3⋊C4, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D4⋊2S3, S3×Q8, C6×Q8, C3×C4○D4, D4.10D4, C12.47D4, D12⋊C4, Q8⋊3Dic3, Q8.11D6, Dic3⋊Q8, C3×C8.C22, Q8○D12, D12.40D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.10D4, C23⋊2D6, D12.40D4
Character table of D12.40D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 24A | 24B | |
| size | 1 | 1 | 2 | 4 | 12 | 2 | 2 | 2 | 4 | 8 | 12 | 12 | 12 | 12 | 12 | 2 | 4 | 8 | 8 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -2 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 2 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | -1 | -√-3 | √-3 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | 2 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 1 | -1 | -√-3 | √-3 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | -2 | 2 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 1 | -1 | √-3 | -√-3 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | -1 | √-3 | -√-3 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
(1 46 10 37 7 40 4 43)(2 41 11 44 8 47 5 38)(3 48 12 39 9 42 6 45)(13 34 22 25 19 28 16 31)(14 29 23 32 20 35 17 26)(15 36 24 27 21 30 18 33)
(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,46,10,37,7,40,4,43)(2,41,11,44,8,47,5,38)(3,48,12,39,9,42,6,45)(13,34,22,25,19,28,16,31)(14,29,23,32,20,35,17,26)(15,36,24,27,21,30,18,33), (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42)>;
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,46,10,37,7,40,4,43)(2,41,11,44,8,47,5,38)(3,48,12,39,9,42,6,45)(13,34,22,25,19,28,16,31)(14,29,23,32,20,35,17,26)(15,36,24,27,21,30,18,33), (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42) );
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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)], [(1,46,10,37,7,40,4,43),(2,41,11,44,8,47,5,38),(3,48,12,39,9,42,6,45),(13,34,22,25,19,28,16,31),(14,29,23,32,20,35,17,26),(15,36,24,27,21,30,18,33)], [(1,25,7,31),(2,26,8,32),(3,27,9,33),(4,28,10,34),(5,29,11,35),(6,30,12,36),(13,43,19,37),(14,44,20,38),(15,45,21,39),(16,46,22,40),(17,47,23,41),(18,48,24,42)]])
Matrix representation of D12.40D4 ►in GL8(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 39 | 72 | 70 |
| 0 | 0 | 0 | 0 | 26 | 0 | 25 | 1 |
| 24 | 49 | 69 | 4 | 0 | 0 | 0 | 0 |
| 25 | 49 | 8 | 4 | 0 | 0 | 0 | 0 |
| 4 | 69 | 49 | 24 | 0 | 0 | 0 | 0 |
| 65 | 69 | 48 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 42 | 31 | 46 | 65 |
| 0 | 0 | 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 47 | 28 | 42 |
| 49 | 25 | 4 | 8 | 0 | 0 | 0 | 0 |
| 48 | 24 | 65 | 69 | 0 | 0 | 0 | 0 |
| 69 | 65 | 24 | 48 | 0 | 0 | 0 | 0 |
| 8 | 4 | 25 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 31 | 46 | 65 |
| 0 | 0 | 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 26 | 0 | 31 |
| 69 | 65 | 24 | 48 | 0 | 0 | 0 | 0 |
| 8 | 4 | 25 | 49 | 0 | 0 | 0 | 0 |
| 49 | 25 | 4 | 8 | 0 | 0 | 0 | 0 |
| 48 | 24 | 65 | 69 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 39 | 34 | 1 | 3 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 28 | 0 | 39 |
G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,34,26,0,0,0,0,72,0,39,0,0,0,0,0,0,0,72,25,0,0,0,0,0,0,70,1],[24,25,4,65,0,0,0,0,49,49,69,69,0,0,0,0,69,8,49,48,0,0,0,0,4,4,24,24,0,0,0,0,0,0,0,0,0,42,46,44,0,0,0,0,0,31,0,47,0,0,0,0,27,46,0,28,0,0,0,0,0,65,0,42],[49,48,69,8,0,0,0,0,25,24,65,4,0,0,0,0,4,65,24,25,0,0,0,0,8,69,48,49,0,0,0,0,0,0,0,0,42,0,27,29,0,0,0,0,31,0,0,26,0,0,0,0,46,46,0,0,0,0,0,0,65,0,0,31],[69,8,49,48,0,0,0,0,65,4,25,24,0,0,0,0,24,25,4,65,0,0,0,0,48,49,8,69,0,0,0,0,0,0,0,0,0,39,72,45,0,0,0,0,0,34,0,28,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,39] >;
D12.40D4 in GAP, Magma, Sage, TeX
D_{12}._{40}D_4 % in TeX
G:=Group("D12.40D4"); // GroupNames label
G:=SmallGroup(192,764);
// by ID
G=gap.SmallGroup(192,764);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,297,136,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^6*c^3>;
// generators/relations
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