metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:5D6, D4.8D12, D12.33D4, M4(2):4D6, Q8.13D12, Dic6.33D4, C4wrC2:2S3, (C3xD4).3D4, C12.4(C2xD4), (C3xQ8).3D4, C4oD4.18D6, D4oD12.1C2, C4.10(C2xD12), C8.D6:8C2, C4.126(S3xD4), C42:4S3:8C2, (C4xC12):12C22, C6.28C22wrC2, Q8.14D6:1C2, C3:2(D4.9D4), (C22xS3).3D4, C22.30(S3xD4), C42:7S3:10C2, C12.46D4:2C2, C4.Dic3:5C22, C2.31(D6:D4), (C2xC12).265C23, (C2xDic6):14C22, C4oD12.14C22, (C2xD12).70C22, (C3xM4(2)):11C22, (C3xC4wrC2):2C2, (C2xC6).27(C2xD4), (C3xC4oD4).6C22, (C2xC4).110(C22xS3), SmallGroup(192,384)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:5D6
G = < a,b,c,d | a4=b4=c6=d2=1, cac-1=ab=ba, dad=a-1b2, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 544 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, M4(2), M4(2), SD16, Q16, C2xD4, C2xQ8, C4oD4, C4oD4, C3:C8, C24, Dic6, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C22xS3, C4.D4, C4wrC2, C4wrC2, C4.4D4, C8.C22, 2+ 1+4, C24:C2, Dic12, C4.Dic3, D6:C4, D4.S3, C3:Q16, C4xC12, C3xM4(2), C2xDic6, C2xD12, C2xD12, C4oD12, C4oD12, S3xD4, Q8:3S3, C3xC4oD4, D4.9D4, C42:4S3, C12.46D4, C3xC4wrC2, C42:7S3, C8.D6, Q8.14D6, D4oD12, C42:5D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C22wrC2, C2xD12, S3xD4, D4.9D4, D6:D4, C42:5D6
Character table of C42:5D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | 24B | |
| size | 1 | 1 | 2 | 4 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 24 | 2 | 4 | 8 | 8 | 24 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 2 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -1 | 1 | -2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 2 | 0 | 2 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 0 | -2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | √3 | -√3 | -1 | √3 | -√3 | 1 | -√3 | √3 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | √3 | -√3 | -1 | √3 | -√3 | -1 | √3 | -√3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | -√3 | √3 | -1 | -√3 | √3 | 1 | √3 | -√3 | orthogonal lifted from D12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | -√3 | √3 | -1 | -√3 | √3 | -1 | -√3 | √3 | orthogonal lifted from D12 |
| ρ23 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from D4.9D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from D4.9D4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | ζ43+2ζ32+1 | ζ4+2ζ32+1 | 0 | ζ4+2ζ3+1 | ζ43+2ζ3+1 | 0 | 0 | 0 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | ζ43+2ζ3+1 | ζ4+2ζ3+1 | 0 | ζ4+2ζ32+1 | ζ43+2ζ32+1 | 0 | 0 | 0 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | ζ4+2ζ3+1 | ζ43+2ζ3+1 | 0 | ζ43+2ζ32+1 | ζ4+2ζ32+1 | 0 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | ζ4+2ζ32+1 | ζ43+2ζ32+1 | 0 | ζ43+2ζ3+1 | ζ4+2ζ3+1 | 0 | 0 | 0 | complex faithful |
►On 48 points
(1 34 26 43)(2 32 30 47)(3 36 28 45)(4 41 25 22)(5 39 29 20)(6 37 27 24)(7 19 18 33)(8 23 16 31)(9 21 14 35)(10 46 13 42)(11 44 17 40)(12 48 15 38)
(1 16 5 13)(2 14 6 17)(3 18 4 15)(7 25 12 28)(8 29 10 26)(9 27 11 30)(19 22 48 45)(20 46 43 23)(21 24 44 47)(31 39 42 34)(32 35 37 40)(33 41 38 36)
(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 5)(2 4)(3 6)(7 9)(11 12)(14 18)(15 17)(19 40)(20 39)(21 38)(22 37)(23 42)(24 41)(25 30)(26 29)(27 28)(31 46)(32 45)(33 44)(34 43)(35 48)(36 47)
G:=sub<Sym(48)| (1,34,26,43)(2,32,30,47)(3,36,28,45)(4,41,25,22)(5,39,29,20)(6,37,27,24)(7,19,18,33)(8,23,16,31)(9,21,14,35)(10,46,13,42)(11,44,17,40)(12,48,15,38), (1,16,5,13)(2,14,6,17)(3,18,4,15)(7,25,12,28)(8,29,10,26)(9,27,11,30)(19,22,48,45)(20,46,43,23)(21,24,44,47)(31,39,42,34)(32,35,37,40)(33,41,38,36), (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), (1,5)(2,4)(3,6)(7,9)(11,12)(14,18)(15,17)(19,40)(20,39)(21,38)(22,37)(23,42)(24,41)(25,30)(26,29)(27,28)(31,46)(32,45)(33,44)(34,43)(35,48)(36,47)>;
G:=Group( (1,34,26,43)(2,32,30,47)(3,36,28,45)(4,41,25,22)(5,39,29,20)(6,37,27,24)(7,19,18,33)(8,23,16,31)(9,21,14,35)(10,46,13,42)(11,44,17,40)(12,48,15,38), (1,16,5,13)(2,14,6,17)(3,18,4,15)(7,25,12,28)(8,29,10,26)(9,27,11,30)(19,22,48,45)(20,46,43,23)(21,24,44,47)(31,39,42,34)(32,35,37,40)(33,41,38,36), (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), (1,5)(2,4)(3,6)(7,9)(11,12)(14,18)(15,17)(19,40)(20,39)(21,38)(22,37)(23,42)(24,41)(25,30)(26,29)(27,28)(31,46)(32,45)(33,44)(34,43)(35,48)(36,47) );
G=PermutationGroup([[(1,34,26,43),(2,32,30,47),(3,36,28,45),(4,41,25,22),(5,39,29,20),(6,37,27,24),(7,19,18,33),(8,23,16,31),(9,21,14,35),(10,46,13,42),(11,44,17,40),(12,48,15,38)], [(1,16,5,13),(2,14,6,17),(3,18,4,15),(7,25,12,28),(8,29,10,26),(9,27,11,30),(19,22,48,45),(20,46,43,23),(21,24,44,47),(31,39,42,34),(32,35,37,40),(33,41,38,36)], [(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)], [(1,5),(2,4),(3,6),(7,9),(11,12),(14,18),(15,17),(19,40),(20,39),(21,38),(22,37),(23,42),(24,41),(25,30),(26,29),(27,28),(31,46),(32,45),(33,44),(34,43),(35,48),(36,47)]])
Matrix representation of C42:5D6 ►in GL4(F73) generated by
| 57 | 13 | 65 | 43 |
| 60 | 70 | 30 | 35 |
| 16 | 60 | 0 | 0 |
| 13 | 3 | 0 | 0 |
| 72 | 0 | 72 | 0 |
| 0 | 72 | 0 | 72 |
| 2 | 0 | 1 | 0 |
| 0 | 2 | 0 | 1 |
| 0 | 72 | 0 | 72 |
| 1 | 72 | 1 | 72 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 72 | 0 | 72 | 0 |
| 72 | 1 | 72 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
G:=sub<GL(4,GF(73))| [57,60,16,13,13,70,60,3,65,30,0,0,43,35,0,0],[72,0,2,0,0,72,0,2,72,0,1,0,0,72,0,1],[0,1,0,0,72,72,0,0,0,1,0,72,72,72,1,1],[72,72,0,0,0,1,0,0,72,72,1,1,0,1,0,72] >;
C42:5D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_5D_6
% in TeX
G:=Group("C4^2:5D6"); // GroupNames label
G:=SmallGroup(192,384);
// by ID
G=gap.SmallGroup(192,384);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,1123,136,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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