metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊7S3, C4.5D12, C12.28D4, (C4×C12)⋊5C2, D6⋊C4⋊1C2, C6.4(C2×D4), (C2×C4).77D6, C2.6(C2×D12), (C2×Dic6)⋊1C2, (C2×D12).2C2, C6.5(C4○D4), C3⋊1(C4.4D4), C2.7(C4○D12), (C2×C6).16C23, (C2×C12).74C22, (C22×S3).2C22, C22.37(C22×S3), (C2×Dic3).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, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4.4D4, D6⋊C4, C4×C12, C2×Dic6, C2×D12, C42⋊7S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C2×D12, C4○D12, 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 C4○D4 |
| 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 C4○D4 |
| 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 C4○D4 |
| 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 C4○D4 |
| 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 C4○D12 |
| 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 C4○D12 |
| 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 C4○D12 |
| 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 C4○D12 |
| 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 C4○D12 |
| 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 C4○D12 |
| 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 C4○D12 |
| 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 C4○D12 |
►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 S3×C4.4D4 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
(C2×C4).17D12 C6.C22≀C2 (C22×S3)⋊Q8 (C2×C4).21D12 C12.14Q16 C4.5D24 C42.264D6 C42.14D6 C42.19D6 C42.20D6 (C2×Dic6)⋊7C4 C42⋊11Dic3 (C2×C4)⋊6D12 (C2×C42)⋊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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