metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.6D4, Dic6.6D4, M4(2).5D6, C3⋊C8.26D4, D12.C4⋊7C2, C4.152(S3×D4), C12.99(C2×D4), C8⋊D6.1C2, (C2×Q8).31D6, C4.10D4⋊3S3, C3⋊2(D4.3D4), (C6×Q8).9C22, Q8.11D6⋊1C2, C12.53D4⋊3C2, C6.11(C4⋊D4), (C2×C12).11C23, C4○D12.7C22, C12.46D4⋊12C2, C2.14(Dic3⋊D4), (C2×D12).42C22, C4.Dic3.6C22, C22.15(C4○D12), (C3×M4(2)).22C22, (C2×C3⋊C8).3C22, (C2×Q8⋊2S3)⋊1C2, (C2×C6).32(C4○D4), (C3×C4.10D4)⋊1C2, (C2×C4).11(C22×S3), SmallGroup(192,313)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.6D4
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=a3c3 >
Subgroups: 336 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3 [×2], C6, C6, C8 [×5], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, Dic3, C12 [×2], C12, D6 [×3], C2×C6, C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8 [×2], C3⋊C8, C24 [×2], Dic6, C4×S3, D12, D12 [×2], C3⋊D4, C2×C12, C2×C12, C3×Q8 [×2], C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2, D24, C2×C3⋊C8, C4.Dic3, Q8⋊2S3 [×3], C3⋊Q16, C3×M4(2) [×2], C2×D12, C4○D12, C6×Q8, D4.3D4, C12.53D4, C12.46D4, C3×C4.10D4, D12.C4, C8⋊D6, C2×Q8⋊2S3, Q8.11D6, D12.6D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4 [×2], D4.3D4, Dic3⋊D4, D12.6D4
Character table of D12.6D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 24 | 2 | 2 | 2 | 8 | 12 | 2 | 4 | 4 | 4 | 6 | 6 | 8 | 12 | 24 | 4 | 4 | 8 | 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 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | -√3 | i | -i | √3 | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | √3 | i | -i | -√3 | complex lifted from C4○D12 |
| ρ21 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | √3 | -i | i | -√3 | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | -√3 | -i | i | √3 | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.3D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.3D4 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►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 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 28 10 31 7 34 4 25)(2 35 11 26 8 29 5 32)(3 30 12 33 9 36 6 27)(13 46 22 37 19 40 16 43)(14 41 23 44 20 47 17 38)(15 48 24 39 21 42 18 45)
(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 34 16 25 19 28 22 31)(14 35 17 26 20 29 23 32)(15 36 18 27 21 30 24 33)
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)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28), (1,28,10,31,7,34,4,25)(2,35,11,26,8,29,5,32)(3,30,12,33,9,36,6,27)(13,46,22,37,19,40,16,43)(14,41,23,44,20,47,17,38)(15,48,24,39,21,42,18,45), (1,46,4,37,7,40,10,43)(2,47,5,38,8,41,11,44)(3,48,6,39,9,42,12,45)(13,34,16,25,19,28,22,31)(14,35,17,26,20,29,23,32)(15,36,18,27,21,30,24,33)>;
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)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28), (1,28,10,31,7,34,4,25)(2,35,11,26,8,29,5,32)(3,30,12,33,9,36,6,27)(13,46,22,37,19,40,16,43)(14,41,23,44,20,47,17,38)(15,48,24,39,21,42,18,45), (1,46,4,37,7,40,10,43)(2,47,5,38,8,41,11,44)(3,48,6,39,9,42,12,45)(13,34,16,25,19,28,22,31)(14,35,17,26,20,29,23,32)(15,36,18,27,21,30,24,33) );
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),(13,27),(14,26),(15,25),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28)], [(1,28,10,31,7,34,4,25),(2,35,11,26,8,29,5,32),(3,30,12,33,9,36,6,27),(13,46,22,37,19,40,16,43),(14,41,23,44,20,47,17,38),(15,48,24,39,21,42,18,45)], [(1,46,4,37,7,40,10,43),(2,47,5,38,8,41,11,44),(3,48,6,39,9,42,12,45),(13,34,16,25,19,28,22,31),(14,35,17,26,20,29,23,32),(15,36,18,27,21,30,24,33)])
Matrix representation of D12.6D4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 67 | 12 | 0 | 72 |
| 0 | 0 | 6 | 0 | 1 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 12 |
| 0 | 0 | 72 | 1 | 0 | 12 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 67 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 12 | 71 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 35 | 2 | 6 | 6 |
| 0 | 0 | 36 | 1 | 6 | 6 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 71 | 61 | 61 |
| 0 | 0 | 1 | 72 | 0 | 61 |
| 0 | 0 | 67 | 12 | 72 | 0 |
| 0 | 0 | 61 | 12 | 72 | 0 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,1,67,6,0,0,71,72,12,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,67,0,0,0,1,0,0,0,0,61,0,72,72,0,0,12,12,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,61,0,35,36,0,0,12,0,2,1,0,0,71,72,6,6,0,0,0,1,6,6],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,2,1,67,61,0,0,71,72,12,12,0,0,61,0,72,72,0,0,61,61,0,0] >;
D12.6D4 in GAP, Magma, Sage, TeX
D_{12}._6D_4 % in TeX
G:=Group("D12.6D4"); // GroupNames label
G:=SmallGroup(192,313);
// by ID
G=gap.SmallGroup(192,313);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,184,297,136,1684,851,6278]);
// 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^3*c^3>;
// generators/relations
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