metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.3D4, Dic6.3D4, M4(2).2D6, C3⋊C8.25D4, C8⋊D6⋊6C2, D12.C4⋊6C2, (C2×D4).16D6, C4.D4⋊4S3, C4.149(S3×D4), C12.94(C2×D4), D12⋊6C22⋊1C2, C3⋊1(D4.4D4), (C2×C12).6C23, C12.53D4⋊2C2, C6.10(C4⋊D4), C4○D12.4C22, (C6×D4).16C22, C12.46D4⋊10C2, C2.13(Dic3⋊D4), (C2×D12).39C22, C4.Dic3.3C22, C22.14(C4○D12), (C3×M4(2)).19C22, (C2×D4⋊S3)⋊1C2, (C2×C3⋊C8).2C22, (C3×C4.D4)⋊2C2, (C2×C4).6(C22×S3), (C2×C6).31(C4○D4), SmallGroup(192,308)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.3D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=a9c3 >
Subgroups: 368 in 108 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4.D4, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, S3×C8, C8⋊S3, C24⋊C2, D24, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3, C3×M4(2), C2×D12, C4○D12, C6×D4, D4.4D4, C12.53D4, C12.46D4, C3×C4.D4, D12.C4, C8⋊D6, C2×D4⋊S3, D12⋊6C22, D12.3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4, D4.4D4, Dic3⋊D4, D12.3D4
Character table of D12.3D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 8 | 12 | 24 | 2 | 2 | 2 | 12 | 2 | 4 | 8 | 8 | 4 | 4 | 6 | 6 | 8 | 12 | 24 | 4 | 4 | 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 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | 1 | 1 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2i | -2i | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | -2i | 2i | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -1 | -1 | √-3 | -√-3 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√3 | -i | i | √3 | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -1 | -1 | -√-3 | √-3 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √3 | -i | i | -√3 | complex lifted from C4○D12 |
| ρ21 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -1 | -1 | -√-3 | √-3 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√3 | i | -i | √3 | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -1 | -1 | √-3 | -√-3 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √3 | i | -i | -√3 | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal 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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(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 34 10 25 7 28 4 31)(2 29 11 32 8 35 5 26)(3 36 12 27 9 30 6 33)(13 37 22 40 19 43 16 46)(14 44 23 47 20 38 17 41)(15 39 24 42 21 45 18 48)
(1 46 4 37 7 40 10 43)(2 47 5 38 8 41 11 44)(3 48 6 39 9 42 12 45)(13 31 16 34 19 25 22 28)(14 32 17 35 20 26 23 29)(15 33 18 36 21 27 24 30)
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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,34,10,25,7,28,4,31)(2,29,11,32,8,35,5,26)(3,36,12,27,9,30,6,33)(13,37,22,40,19,43,16,46)(14,44,23,47,20,38,17,41)(15,39,24,42,21,45,18,48), (1,46,4,37,7,40,10,43)(2,47,5,38,8,41,11,44)(3,48,6,39,9,42,12,45)(13,31,16,34,19,25,22,28)(14,32,17,35,20,26,23,29)(15,33,18,36,21,27,24,30)>;
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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,34,10,25,7,28,4,31)(2,29,11,32,8,35,5,26)(3,36,12,27,9,30,6,33)(13,37,22,40,19,43,16,46)(14,44,23,47,20,38,17,41)(15,39,24,42,21,45,18,48), (1,46,4,37,7,40,10,43)(2,47,5,38,8,41,11,44)(3,48,6,39,9,42,12,45)(13,31,16,34,19,25,22,28)(14,32,17,35,20,26,23,29)(15,33,18,36,21,27,24,30) );
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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)], [(1,34,10,25,7,28,4,31),(2,29,11,32,8,35,5,26),(3,36,12,27,9,30,6,33),(13,37,22,40,19,43,16,46),(14,44,23,47,20,38,17,41),(15,39,24,42,21,45,18,48)], [(1,46,4,37,7,40,10,43),(2,47,5,38,8,41,11,44),(3,48,6,39,9,42,12,45),(13,31,16,34,19,25,22,28),(14,32,17,35,20,26,23,29),(15,33,18,36,21,27,24,30)]])
Matrix representation of D12.3D4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 65 | 65 | 72 | 3 |
| 0 | 0 | 46 | 0 | 48 | 1 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 0 | 70 | 41 |
| 0 | 0 | 6 | 0 | 38 | 0 |
| 0 | 0 | 48 | 25 | 12 | 55 |
| 0 | 0 | 57 | 41 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 48 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 70 | 70 | 27 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 3 | 32 |
| 0 | 0 | 67 | 0 | 35 | 41 |
| 0 | 0 | 25 | 48 | 61 | 18 |
| 0 | 0 | 57 | 0 | 0 | 0 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,72,65,46,0,0,2,72,65,0,0,0,0,0,72,48,0,0,0,0,3,1],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,61,6,48,57,0,0,0,0,25,41,0,0,70,38,12,0,0,0,41,0,55,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,70,72,0,0,0,0,70,72,0,0,48,0,27,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,67,25,57,0,0,0,0,48,0,0,0,3,35,61,0,0,0,32,41,18,0] >;
D12.3D4 in GAP, Magma, Sage, TeX
D_{12}._3D_4 % in TeX
G:=Group("D12.3D4"); // GroupNames label
G:=SmallGroup(192,308);
// by ID
G=gap.SmallGroup(192,308);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,297,136,1684,851,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^3>;
// generators/relations
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