metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.2D4, Dic6.2D4, M4(2).1D6, C3⋊C8.24D4, D12.C4⋊5C2, (C2×D4).15D6, C4.D4⋊3S3, C8.D6⋊6C2, C4.148(S3×D4), C12.93(C2×D4), C6.9(C4⋊D4), C3⋊1(D4.3D4), (C2×C12).5C23, C12.53D4⋊1C2, D12⋊6C22.1C2, C4○D12.3C22, (C6×D4).15C22, C12.47D4⋊10C2, C2.12(Dic3⋊D4), C4.Dic3.2C22, C22.13(C4○D12), (C2×Dic6).45C22, (C3×M4(2)).18C22, (C2×D4.S3)⋊1C2, (C2×C3⋊C8).1C22, (C3×C4.D4)⋊1C2, (C2×C4).5(C22×S3), (C2×C6).30(C4○D4), SmallGroup(192,307)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.2D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=a3c3 >
Subgroups: 304 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, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×C6, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2, Dic12, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3, C3×M4(2), C2×Dic6, C4○D12, C6×D4, D4.3D4, C12.53D4, C12.47D4, C3×C4.D4, D12.C4, C8.D6, D12⋊6C22, C2×D4.S3, D12.2D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4, D4.3D4, Dic3⋊D4, D12.2D4
Character table of D12.2D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 8 | 12 | 2 | 2 | 2 | 12 | 24 | 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 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 2 | -2 | 0 | 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 | -1 | 2 | 2 | 0 | 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 | 0 | -2 | 2 | -2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 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 | 2 | -2 | -2 | 0 | 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 | 2 | -2 | -2 | 0 | 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 | -1 | -2 | -2 | 0 | 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 | -1 | -2 | -2 | 0 | 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 | -1 | -2 | -2 | 0 | 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 | -1 | -2 | -2 | 0 | 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 | -2 | -4 | 4 | 0 | 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 | -2 | 4 | -4 | 0 | 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 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 2√-2 | -2√-2 | 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 | 0 | 0 | -2√-2 | 2√-2 | 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 | 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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 36)(9 35)(10 34)(11 33)(12 32)(13 38)(14 37)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)
(1 22 10 13 7 16 4 19)(2 17 11 20 8 23 5 14)(3 24 12 15 9 18 6 21)(25 41 28 38 31 47 34 44)(26 48 29 45 32 42 35 39)(27 43 30 40 33 37 36 46)
(1 16 10 13 7 22 4 19)(2 17 11 14 8 23 5 20)(3 18 12 15 9 24 6 21)(25 44 34 41 31 38 28 47)(26 45 35 42 32 39 29 48)(27 46 36 43 33 40 30 37)
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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,36)(9,35)(10,34)(11,33)(12,32)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39), (1,22,10,13,7,16,4,19)(2,17,11,20,8,23,5,14)(3,24,12,15,9,18,6,21)(25,41,28,38,31,47,34,44)(26,48,29,45,32,42,35,39)(27,43,30,40,33,37,36,46), (1,16,10,13,7,22,4,19)(2,17,11,14,8,23,5,20)(3,18,12,15,9,24,6,21)(25,44,34,41,31,38,28,47)(26,45,35,42,32,39,29,48)(27,46,36,43,33,40,30,37)>;
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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,36)(9,35)(10,34)(11,33)(12,32)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39), (1,22,10,13,7,16,4,19)(2,17,11,20,8,23,5,14)(3,24,12,15,9,18,6,21)(25,41,28,38,31,47,34,44)(26,48,29,45,32,42,35,39)(27,43,30,40,33,37,36,46), (1,16,10,13,7,22,4,19)(2,17,11,14,8,23,5,20)(3,18,12,15,9,24,6,21)(25,44,34,41,31,38,28,47)(26,45,35,42,32,39,29,48)(27,46,36,43,33,40,30,37) );
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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,36),(9,35),(10,34),(11,33),(12,32),(13,38),(14,37),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39)], [(1,22,10,13,7,16,4,19),(2,17,11,20,8,23,5,14),(3,24,12,15,9,18,6,21),(25,41,28,38,31,47,34,44),(26,48,29,45,32,42,35,39),(27,43,30,40,33,37,36,46)], [(1,16,10,13,7,22,4,19),(2,17,11,14,8,23,5,20),(3,18,12,15,9,24,6,21),(25,44,34,41,31,38,28,47),(26,45,35,42,32,39,29,48),(27,46,36,43,33,40,30,37)]])
Matrix representation of D12.2D4 ►in GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 71 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 29 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 22 | 45 | 1 | 63 |
| 0 | 0 | 45 | 29 | 44 | 72 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 2 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 45 | 56 | 12 |
| 0 | 0 | 59 | 0 | 12 | 0 |
| 0 | 0 | 13 | 10 | 41 | 14 |
| 0 | 0 | 10 | 28 | 28 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 29 | 0 |
| 0 | 0 | 0 | 1 | 0 | 10 |
| 0 | 0 | 0 | 1 | 46 | 0 |
| 0 | 0 | 1 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 29 | 0 |
| 0 | 0 | 22 | 72 | 2 | 63 |
| 0 | 0 | 0 | 1 | 46 | 0 |
| 0 | 0 | 27 | 44 | 29 | 1 |
G:=sub<GL(6,GF(73))| [64,71,0,0,0,0,0,8,0,0,0,0,0,0,72,10,22,45,0,0,29,1,45,29,0,0,0,0,1,44,0,0,0,0,63,72],[8,2,0,0,0,0,5,65,0,0,0,0,0,0,32,59,13,10,0,0,45,0,10,28,0,0,56,12,41,28,0,0,12,0,14,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,1,0,0,0,1,1,0,0,0,29,0,46,0,0,0,0,10,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,22,0,27,0,0,0,72,1,44,0,0,29,2,46,29,0,0,0,63,0,1] >;
D12.2D4 in GAP, Magma, Sage, TeX
D_{12}._2D_4 % in TeX
G:=Group("D12.2D4"); // GroupNames label
G:=SmallGroup(192,307);
// by ID
G=gap.SmallGroup(192,307);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,297,136,1684,851,6278]);
// 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^6*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^3>;
// generators/relations
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