metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.5D4, Dic6.5D4, M4(2).4D6, C8⋊D6⋊7C2, (C2×C4).7D12, (C2×C12).9D4, C4.81(S3×D4), C12.98(C2×D4), (C2×Q8).30D6, D12⋊C4⋊4C2, C6.18C22≀C2, C4.10D4⋊2S3, C3⋊1(D4.8D4), (C6×Q8).8C22, C12.23D4⋊1C2, (C2×C12).10C23, Q8.15D6⋊1C2, C4○D12.6C22, C22.13(C2×D12), C2.21(D6⋊D4), (C2×D12).41C22, (C4×Dic3).2C22, (C3×M4(2)).3C22, (C2×C6).23(C2×D4), (C3×C4.10D4)⋊4C2, (C2×C4).10(C22×S3), SmallGroup(192,312)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.5D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, dbd-1=a9b, dcd-1=a9c3 >
Subgroups: 496 in 146 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C24⋊C2, D24, C4×Dic3, D6⋊C4, C3×M4(2), C2×D12, C4○D12, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, D4.8D4, D12⋊C4, C3×C4.10D4, C8⋊D6, C12.23D4, Q8.15D6, D12.5D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4.8D4, D6⋊D4, D12.5D4
Character table of D12.5D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 8A | 8B | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 4 | 8 | 8 | 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 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -2 | 2 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | -1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | -1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.8D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.8D4 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)
(1 16 10 19 7 22 4 13)(2 23 11 14 8 17 5 20)(3 18 12 21 9 24 6 15)(25 43 34 46 31 37 28 40)(26 38 35 41 32 44 29 47)(27 45 36 48 33 39 30 42)
(1 46 10 43 7 40 4 37)(2 47 11 44 8 41 5 38)(3 48 12 45 9 42 6 39)(13 34 22 31 19 28 16 25)(14 35 23 32 20 29 17 26)(15 36 24 33 21 30 18 27)
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,16,10,19,7,22,4,13)(2,23,11,14,8,17,5,20)(3,18,12,21,9,24,6,15)(25,43,34,46,31,37,28,40)(26,38,35,41,32,44,29,47)(27,45,36,48,33,39,30,42), (1,46,10,43,7,40,4,37)(2,47,11,44,8,41,5,38)(3,48,12,45,9,42,6,39)(13,34,22,31,19,28,16,25)(14,35,23,32,20,29,17,26)(15,36,24,33,21,30,18,27)>;
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,16,10,19,7,22,4,13)(2,23,11,14,8,17,5,20)(3,18,12,21,9,24,6,15)(25,43,34,46,31,37,28,40)(26,38,35,41,32,44,29,47)(27,45,36,48,33,39,30,42), (1,46,10,43,7,40,4,37)(2,47,11,44,8,41,5,38)(3,48,12,45,9,42,6,39)(13,34,22,31,19,28,16,25)(14,35,23,32,20,29,17,26)(15,36,24,33,21,30,18,27) );
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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46)], [(1,16,10,19,7,22,4,13),(2,23,11,14,8,17,5,20),(3,18,12,21,9,24,6,15),(25,43,34,46,31,37,28,40),(26,38,35,41,32,44,29,47),(27,45,36,48,33,39,30,42)], [(1,46,10,43,7,40,4,37),(2,47,11,44,8,41,5,38),(3,48,12,45,9,42,6,39),(13,34,22,31,19,28,16,25),(14,35,23,32,20,29,17,26),(15,36,24,33,21,30,18,27)]])
Matrix representation of D12.5D4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 65 | 56 |
| 0 | 0 | 0 | 27 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 60 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 27 |
| 0 | 0 | 46 | 0 | 65 | 56 |
| 0 | 0 | 0 | 0 | 56 | 37 |
| 0 | 0 | 0 | 0 | 8 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 51 |
| 0 | 0 | 0 | 0 | 65 | 56 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 27 | 0 | 8 | 17 |
| 0 | 0 | 60 | 25 | 17 | 3 |
| 0 | 0 | 19 | 0 | 65 | 56 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,65,0,27,0,0,0,56,27,60,46],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,65,56,8,0,0,27,56,37,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,1,0,0,0,0,0,0,0,17,65,0,0,1,0,51,56],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,60,19,0,0,72,0,25,0,0,0,0,8,17,65,0,0,0,17,3,56] >;
D12.5D4 in GAP, Magma, Sage, TeX
D_{12}._5D_4 % in TeX
G:=Group("D12.5D4"); // GroupNames label
G:=SmallGroup(192,312);
// by ID
G=gap.SmallGroup(192,312);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,226,1123,570,136,1684,438,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=a^9*c^3>;
// generators/relations
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