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