metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:7S3, C4.5D12, C12.28D4, (C4xC12):5C2, D6:C4:1C2, C6.4(C2xD4), (C2xC4).77D6, C2.6(C2xD12), (C2xDic6):1C2, (C2xD12).2C2, C6.5(C4oD4), C3:1(C4.4D4), C2.7(C4oD12), (C2xC6).16C23, (C2xC12).74C22, (C22xS3).2C22, C22.37(C22xS3), (C2xDic3).3C22, SmallGroup(96,82)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:7S3
G = < a,b,c,d | a4=b4=c3=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 202 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, C2xC6, C42, C22:C4, C2xD4, C2xQ8, Dic6, D12, C2xDic3, C2xC12, C2xC12, C22xS3, C4.4D4, D6:C4, C4xC12, C2xDic6, C2xD12, C42:7S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, D12, C22xS3, C4.4D4, C2xD12, C4oD12, C42:7S3
Character table of C42:7S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | -1 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | √3 | -√3 | 1 | 1 | √3 | -1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | -√3 | √3 | 1 | 1 | -√3 | -1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | √3 | √3 | -1 | -1 | -√3 | 1 | 1 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | -√3 | -√3 | -1 | -1 | √3 | 1 | 1 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2i | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 2i | complex lifted from C4oD4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | 0 | complex lifted from C4oD4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2i | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | -2i | complex lifted from C4oD4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | -i | √3 | -√3 | -i | -√3 | √3 | i | √-3 | -√-3 | i | complex lifted from C4oD12 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -i | -i | -√-3 | -√3 | √3 | √-3 | -√3 | √3 | √-3 | i | i | -√-3 | complex lifted from C4oD12 |
| ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -i | -√3 | √3 | -i | √3 | -√3 | i | -√-3 | √-3 | i | complex lifted from C4oD12 |
| ρ26 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | i | -√3 | √3 | i | √3 | -√3 | -i | √-3 | -√-3 | -i | complex lifted from C4oD12 |
| ρ27 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | i | √3 | -√3 | i | -√3 | √3 | -i | -√-3 | √-3 | -i | complex lifted from C4oD12 |
| ρ28 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -i | -i | √-3 | √3 | -√3 | -√-3 | √3 | -√3 | -√-3 | i | i | √-3 | complex lifted from C4oD12 |
| ρ29 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 1 | 1 | i | i | -√-3 | √3 | -√3 | √-3 | √3 | -√3 | √-3 | -i | -i | -√-3 | complex lifted from C4oD12 |
| ρ30 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 1 | 1 | i | i | √-3 | -√3 | √3 | -√-3 | -√3 | √3 | -√-3 | -i | -i | √-3 | complex lifted from C4oD12 |
►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 26 43 17)(2 27 44 18)(3 28 41 19)(4 25 42 20)(5 22 34 16)(6 23 35 13)(7 24 36 14)(8 21 33 15)(9 30 47 39)(10 31 48 40)(11 32 45 37)(12 29 46 38)
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 38 20)(6 39 17)(7 40 18)(8 37 19)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(25 34 29)(26 35 30)(27 36 31)(28 33 32)
(2 44)(4 42)(5 31)(6 37)(7 29)(8 39)(9 23)(10 14)(11 21)(12 16)(13 47)(15 45)(17 19)(18 25)(20 27)(22 46)(24 48)(26 28)(30 33)(32 35)(34 40)(36 38)
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), (1,26,43,17)(2,27,44,18)(3,28,41,19)(4,25,42,20)(5,22,34,16)(6,23,35,13)(7,24,36,14)(8,21,33,15)(9,30,47,39)(10,31,48,40)(11,32,45,37)(12,29,46,38), (1,23,9)(2,24,10)(3,21,11)(4,22,12)(5,38,20)(6,39,17)(7,40,18)(8,37,19)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(25,34,29)(26,35,30)(27,36,31)(28,33,32), (2,44)(4,42)(5,31)(6,37)(7,29)(8,39)(9,23)(10,14)(11,21)(12,16)(13,47)(15,45)(17,19)(18,25)(20,27)(22,46)(24,48)(26,28)(30,33)(32,35)(34,40)(36,38)>;
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), (1,26,43,17)(2,27,44,18)(3,28,41,19)(4,25,42,20)(5,22,34,16)(6,23,35,13)(7,24,36,14)(8,21,33,15)(9,30,47,39)(10,31,48,40)(11,32,45,37)(12,29,46,38), (1,23,9)(2,24,10)(3,21,11)(4,22,12)(5,38,20)(6,39,17)(7,40,18)(8,37,19)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(25,34,29)(26,35,30)(27,36,31)(28,33,32), (2,44)(4,42)(5,31)(6,37)(7,29)(8,39)(9,23)(10,14)(11,21)(12,16)(13,47)(15,45)(17,19)(18,25)(20,27)(22,46)(24,48)(26,28)(30,33)(32,35)(34,40)(36,38) );
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)], [(1,26,43,17),(2,27,44,18),(3,28,41,19),(4,25,42,20),(5,22,34,16),(6,23,35,13),(7,24,36,14),(8,21,33,15),(9,30,47,39),(10,31,48,40),(11,32,45,37),(12,29,46,38)], [(1,23,9),(2,24,10),(3,21,11),(4,22,12),(5,38,20),(6,39,17),(7,40,18),(8,37,19),(13,47,43),(14,48,44),(15,45,41),(16,46,42),(25,34,29),(26,35,30),(27,36,31),(28,33,32)], [(2,44),(4,42),(5,31),(6,37),(7,29),(8,39),(9,23),(10,14),(11,21),(12,16),(13,47),(15,45),(17,19),(18,25),(20,27),(22,46),(24,48),(26,28),(30,33),(32,35),(34,40),(36,38)]])
C42:7S3 is a maximal subgroup of
C42.D6 C8.8D12 C42.264D6 C8:D12 C42.20D6 C8.D12 C42:5D6 D4.10D12 D12.19D4 C42.36D6 D4.1D12 Q8.6D12 C42.214D6 C42.216D6 C42.74D6 C42.80D6 C42.82D6 C42.276D6 C42.277D6 C42:11D6 C42.92D6 C42:12D6 C42.97D6 C42.99D6 D12:23D4 Dic6:23D4 D4:5D12 C42.114D6 C42:19D6 C42.122D6 Q8:6D12 C42.133D6 C42.135D6 C42.136D6 C42.233D6 S3xC4.4D4 C42:24D6 C42.145D6 C42.237D6 C42.157D6 C42.158D6 C42:25D6 C42.164D6 C42:28D6 C42.171D6 C42.178D6 C42:7D9 Dic3.D12 C12.27D12 C12.28D12 C122:6C2 Dic5.8D12 C60.69D4 C60.70D4 C42:7D15
C42:7S3 is a maximal quotient of
(C2xC4).17D12 C6.C22wrC2 (C22xS3):Q8 (C2xC4).21D12 C12.14Q16 C4.5D24 C42.264D6 C42.14D6 C42.19D6 C42.20D6 (C2xDic6):7C4 C42:11Dic3 (C2xC4):6D12 (C2xC42):3S3 C42:7D9 Dic3.D12 C12.27D12 C12.28D12 C122:6C2 Dic5.8D12 C60.69D4 C60.70D4 C42:7D15
Matrix representation of C42:7S3 ►in GL4(F13) generated by
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 4 | 2 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(13))| [3,7,0,0,6,10,0,0,0,0,8,0,0,0,0,8],[3,7,0,0,6,10,0,0,0,0,11,4,0,0,9,2],[12,1,0,0,12,0,0,0,0,0,12,1,0,0,12,0],[12,1,0,0,0,1,0,0,0,0,1,12,0,0,0,12] >;
C42:7S3 in GAP, Magma, Sage, TeX
C_4^2\rtimes_7S_3
% in TeX
G:=Group("C4^2:7S3"); // GroupNames label
G:=SmallGroup(96,82);
// by ID
G=gap.SmallGroup(96,82);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,218,86,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^3=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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