direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary
Aliases: C2×S3×Q8, C6.8C24, C12.22C23, Dic6⋊9C22, D6.10C23, Dic3.5C23, C6⋊2(C2×Q8), (C6×Q8)⋊5C2, C3⋊2(C22×Q8), (C2×C4).61D6, C2.9(S3×C23), (C3×Q8)⋊5C22, (C2×Dic6)⋊13C2, C4.22(C22×S3), (C2×C6).66C23, (C4×S3).13C22, (C2×C12).46C22, C22.31(C22×S3), (C22×S3).36C22, (C2×Dic3).44C22, (S3×C2×C4).6C2, SmallGroup(96,212)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×S3×Q8
G = < a,b,c,d,e | a2=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 274 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C22×C4, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C22×S3, C22×Q8, C2×Dic6, S3×C2×C4, S3×Q8, C6×Q8, C2×S3×Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C22×S3, C22×Q8, S3×Q8, S3×C23, C2×S3×Q8
Character table of C2×S3×Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 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 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ22 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ26 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ27 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ28 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
| ρ30 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
►On 48 points
(1 35)(2 36)(3 33)(4 34)(5 29)(6 30)(7 31)(8 32)(9 38)(10 39)(11 40)(12 37)(13 25)(14 26)(15 27)(16 28)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 21 27)(18 22 28)(19 23 25)(20 24 26)(29 35 38)(30 36 39)(31 33 40)(32 34 37)
(1 35)(2 36)(3 33)(4 34)(5 38)(6 39)(7 40)(8 37)(9 29)(10 30)(11 31)(12 32)(13 23)(14 24)(15 21)(16 22)(17 41)(18 42)(19 43)(20 44)(25 47)(26 48)(27 45)(28 46)
(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 43 3 41)(2 42 4 44)(5 13 7 15)(6 16 8 14)(9 47 11 45)(10 46 12 48)(17 35 19 33)(18 34 20 36)(21 38 23 40)(22 37 24 39)(25 31 27 29)(26 30 28 32)
G:=sub<Sym(48)| (1,35)(2,36)(3,33)(4,34)(5,29)(6,30)(7,31)(8,32)(9,38)(10,39)(11,40)(12,37)(13,25)(14,26)(15,27)(16,28)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,43,47)(14,44,48)(15,41,45)(16,42,46)(17,21,27)(18,22,28)(19,23,25)(20,24,26)(29,35,38)(30,36,39)(31,33,40)(32,34,37), (1,35)(2,36)(3,33)(4,34)(5,38)(6,39)(7,40)(8,37)(9,29)(10,30)(11,31)(12,32)(13,23)(14,24)(15,21)(16,22)(17,41)(18,42)(19,43)(20,44)(25,47)(26,48)(27,45)(28,46), (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), (1,43,3,41)(2,42,4,44)(5,13,7,15)(6,16,8,14)(9,47,11,45)(10,46,12,48)(17,35,19,33)(18,34,20,36)(21,38,23,40)(22,37,24,39)(25,31,27,29)(26,30,28,32)>;
G:=Group( (1,35)(2,36)(3,33)(4,34)(5,29)(6,30)(7,31)(8,32)(9,38)(10,39)(11,40)(12,37)(13,25)(14,26)(15,27)(16,28)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,43,47)(14,44,48)(15,41,45)(16,42,46)(17,21,27)(18,22,28)(19,23,25)(20,24,26)(29,35,38)(30,36,39)(31,33,40)(32,34,37), (1,35)(2,36)(3,33)(4,34)(5,38)(6,39)(7,40)(8,37)(9,29)(10,30)(11,31)(12,32)(13,23)(14,24)(15,21)(16,22)(17,41)(18,42)(19,43)(20,44)(25,47)(26,48)(27,45)(28,46), (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), (1,43,3,41)(2,42,4,44)(5,13,7,15)(6,16,8,14)(9,47,11,45)(10,46,12,48)(17,35,19,33)(18,34,20,36)(21,38,23,40)(22,37,24,39)(25,31,27,29)(26,30,28,32) );
G=PermutationGroup([[(1,35),(2,36),(3,33),(4,34),(5,29),(6,30),(7,31),(8,32),(9,38),(10,39),(11,40),(12,37),(13,25),(14,26),(15,27),(16,28),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,43,47),(14,44,48),(15,41,45),(16,42,46),(17,21,27),(18,22,28),(19,23,25),(20,24,26),(29,35,38),(30,36,39),(31,33,40),(32,34,37)], [(1,35),(2,36),(3,33),(4,34),(5,38),(6,39),(7,40),(8,37),(9,29),(10,30),(11,31),(12,32),(13,23),(14,24),(15,21),(16,22),(17,41),(18,42),(19,43),(20,44),(25,47),(26,48),(27,45),(28,46)], [(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)], [(1,43,3,41),(2,42,4,44),(5,13,7,15),(6,16,8,14),(9,47,11,45),(10,46,12,48),(17,35,19,33),(18,34,20,36),(21,38,23,40),(22,37,24,39),(25,31,27,29),(26,30,28,32)]])
C2×S3×Q8 is a maximal subgroup of
(S3×Q8)⋊C4 Q8⋊3D12 D6⋊Q16 D6⋊8SD16 D6⋊5Q16 C42.125D6 Q8⋊6D12 C6.162- 1+4 Dic6⋊21D4 Dic6⋊22D4 C42.141D6 Dic6⋊10D4 C42.171D6 D12⋊8Q8 C6.1072- 1+4
C2×S3×Q8 is a maximal quotient of
C6.102+ 1+4 Dic6⋊10Q8 C42.232D6 D12⋊10Q8 (Q8×Dic3)⋊C2 C6.752- 1+4 Dic6⋊21D4 C6.512+ 1+4 C6.1182+ 1+4 C6.522+ 1+4 Dic6⋊7Q8 C42.236D6 C42.148D6 D12⋊7Q8 Dic6⋊8Q8 Dic6⋊9Q8 D12⋊8Q8 C42.241D6 C42.174D6 D12⋊9Q8
Matrix representation of C2×S3×Q8 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 8 |
| 0 | 0 | 3 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 11 | 8 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,3,0,0,8,1],[1,0,0,0,0,1,0,0,0,0,5,11,0,0,0,8] >;
C2×S3×Q8 in GAP, Magma, Sage, TeX
C_2\times S_3\times Q_8
% in TeX
G:=Group("C2xS3xQ8"); // GroupNames label
G:=SmallGroup(96,212);
// by ID
G=gap.SmallGroup(96,212);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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