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## G = C2×S3×Q8order 96 = 25·3

### Direct product of C2, S3 and Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×S3×Q8
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — C2×S3×Q8
 Lower central C3 — C6 — C2×S3×Q8
 Upper central C1 — C22 — C2×Q8

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

Smallest permutation representation of C2×S3×Q8
On 48 points
Generators in S48
(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)]])

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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