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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, 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, C3⋊Q16, C3×M4(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, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4, 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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 36)(23 35)(24 34)
(1 31 10 34 7 25 4 28)(2 26 11 29 8 32 5 35)(3 33 12 36 9 27 6 30)(13 40 22 43 19 46 16 37)(14 47 23 38 20 41 17 44)(15 42 24 45 21 48 18 39)
(1 37 4 40 7 43 10 46)(2 38 5 41 8 44 11 47)(3 39 6 42 9 45 12 48)(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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,36)(23,35)(24,34), (1,31,10,34,7,25,4,28)(2,26,11,29,8,32,5,35)(3,33,12,36,9,27,6,30)(13,40,22,43,19,46,16,37)(14,47,23,38,20,41,17,44)(15,42,24,45,21,48,18,39), (1,37,4,40,7,43,10,46)(2,38,5,41,8,44,11,47)(3,39,6,42,9,45,12,48)(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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,36)(23,35)(24,34), (1,31,10,34,7,25,4,28)(2,26,11,29,8,32,5,35)(3,33,12,36,9,27,6,30)(13,40,22,43,19,46,16,37)(14,47,23,38,20,41,17,44)(15,42,24,45,21,48,18,39), (1,37,4,40,7,43,10,46)(2,38,5,41,8,44,11,47)(3,39,6,42,9,45,12,48)(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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,48),(11,47),(12,46),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,36),(23,35),(24,34)], [(1,31,10,34,7,25,4,28),(2,26,11,29,8,32,5,35),(3,33,12,36,9,27,6,30),(13,40,22,43,19,46,16,37),(14,47,23,38,20,41,17,44),(15,42,24,45,21,48,18,39)], [(1,37,4,40,7,43,10,46),(2,38,5,41,8,44,11,47),(3,39,6,42,9,45,12,48),(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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