metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.C4, C8.12D6, Dic6.C4, M4(2)⋊5S3, C24.13C22, C12.39C23, C3⋊D4.C4, (S3×C8)⋊8C2, C3⋊2(C8○D4), C4.5(C4×S3), C8⋊S3⋊6C2, D6.2(C2×C4), (C2×C4).46D6, M4(2)○(C3⋊C8), C12.13(C2×C4), C4○D12.3C2, C3⋊C8.12C22, C22.1(C4×S3), (C3×M4(2))⋊6C2, C4.39(C22×S3), C6.16(C22×C4), Dic3.4(C2×C4), (C4×S3).16C22, (C2×C12).26C22, (C2×C3⋊C8)⋊3C2, C2.17(S3×C2×C4), (C2×C6).6(C2×C4), SmallGroup(96,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.C4
G = < a,b,c | a12=b2=1, c4=a6, bab=a-1, cac-1=a7, cbc-1=a6b >
Subgroups: 114 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C2×C8, M4(2), M4(2), C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, S3×C8, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, D12.C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, C8○D4, S3×C2×C4, D12.C4
Character table of D12.C4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 2 | 4 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | -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 | -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 | 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 | 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 | -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 | -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 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | i | i | -i | -i | i | -i | -1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | i | i | -i | -i | i | -i | -1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -i | -i | i | i | i | i | -i | -i | -i | i | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | i | i | -i | -i | i | i | -i | -i | -i | i | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -i | -i | i | i | -i | i | -1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | -i | -i | i | i | -i | i | -1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -i | -i | i | i | -i | -i | i | i | i | -i | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | i | i | -i | -i | -i | -i | i | i | i | -i | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | 1 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ20 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | 1 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | 1 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | i | -i | -i | i | complex lifted from C4×S3 |
| ρ23 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -i | i | i | -i | complex lifted from C4×S3 |
| ρ24 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | 1 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ25 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2ζ87 | 2ζ83 | 2ζ85 | 2ζ8 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ26 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2ζ85 | 2ζ8 | 2ζ87 | 2ζ83 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ27 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2ζ8 | 2ζ85 | 2ζ83 | 2ζ87 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ28 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2ζ83 | 2ζ87 | 2ζ8 | 2ζ85 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex 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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 36)(33 35)(38 48)(39 47)(40 46)(41 45)(42 44)
(1 39 13 36 7 45 19 30)(2 46 14 31 8 40 20 25)(3 41 15 26 9 47 21 32)(4 48 16 33 10 42 22 27)(5 43 17 28 11 37 23 34)(6 38 18 35 12 44 24 29)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,36)(33,35)(38,48)(39,47)(40,46)(41,45)(42,44), (1,39,13,36,7,45,19,30)(2,46,14,31,8,40,20,25)(3,41,15,26,9,47,21,32)(4,48,16,33,10,42,22,27)(5,43,17,28,11,37,23,34)(6,38,18,35,12,44,24,29)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,36)(33,35)(38,48)(39,47)(40,46)(41,45)(42,44), (1,39,13,36,7,45,19,30)(2,46,14,31,8,40,20,25)(3,41,15,26,9,47,21,32)(4,48,16,33,10,42,22,27)(5,43,17,28,11,37,23,34)(6,38,18,35,12,44,24,29) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,15),(16,24),(17,23),(18,22),(19,21),(25,31),(26,30),(27,29),(32,36),(33,35),(38,48),(39,47),(40,46),(41,45),(42,44)], [(1,39,13,36,7,45,19,30),(2,46,14,31,8,40,20,25),(3,41,15,26,9,47,21,32),(4,48,16,33,10,42,22,27),(5,43,17,28,11,37,23,34),(6,38,18,35,12,44,24,29)]])
D12.C4 is a maximal subgroup of
D12.2D4 D12.3D4 D12.6D4 D12.7D4 M4(2).22D6 C42.196D6 D24⋊10C4 D24⋊7C4 M4(2)⋊26D6 S3×C8○D4 M4(2)⋊28D6 D8⋊5D6 D8⋊6D6 C24.C23 SD16.D6 D36.C4 C24.64D6 C24.D6 D12.Dic3 C3⋊C8.22D6 C24.47D6 C40.55D6 C40.35D6 D12.Dic5 D60.4C4 D60.3C4 D12.F5 Dic6.F5 D15⋊C8⋊C2
D12.C4 is a maximal quotient of
C24⋊Q8 C8⋊9D12 D6.4C42 C42.185D6 C24⋊C4⋊C2 C3⋊D4⋊C8 D6⋊C8⋊C2 C3⋊C8⋊26D4 Dic6⋊C8 C42.198D6 D12⋊C8 C12⋊2M4(2) C42.30D6 C42.31D6 C12.88(C2×Q8) C12.7C42 C24⋊D4 C24⋊21D4 D6⋊C8⋊40C2 D36.C4 C24.64D6 C24.D6 D12.Dic3 C3⋊C8.22D6 C24.47D6 C40.55D6 C40.35D6 D12.Dic5 D60.4C4 D60.3C4 D12.F5 Dic6.F5 D15⋊C8⋊C2
Matrix representation of D12.C4 ►in GL4(𝔽5) generated by
| 0 | 0 | 2 | 2 |
| 2 | 0 | 2 | 4 |
| 1 | 2 | 1 | 0 |
| 2 | 0 | 0 | 4 |
| 1 | 4 | 2 | 0 |
| 0 | 3 | 1 | 0 |
| 0 | 2 | 2 | 0 |
| 3 | 0 | 4 | 4 |
| 3 | 1 | 2 | 4 |
| 1 | 0 | 3 | 4 |
| 4 | 0 | 2 | 2 |
| 0 | 3 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,2,1,2,0,0,2,0,2,2,1,0,2,4,0,4],[1,0,0,3,4,3,2,0,2,1,2,4,0,0,0,4],[3,1,4,0,1,0,0,3,2,3,2,3,4,4,2,0] >;
D12.C4 in GAP, Magma, Sage, TeX
D_{12}.C_4 % in TeX
G:=Group("D12.C4"); // GroupNames label
G:=SmallGroup(96,114);
// by ID
G=gap.SmallGroup(96,114);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,188,50,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^2=1,c^4=a^6,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^6*b>;
// generators/relations
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