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## G = C2×Q8⋊3S3order 96 = 25·3

### Direct product of C2 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×Q8⋊3S3
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — C2×Q8⋊3S3
 Lower central C3 — C6 — C2×Q8⋊3S3
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q83S3
G = < a,b,c,d,e | a2=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 338 in 164 conjugacy classes, 89 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C2×C4○D4, S3×C2×C4, C2×D12, Q83S3, C6×Q8, C2×Q83S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, Q83S3, S3×C23, C2×Q83S3

Character table of C2×Q83S3

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 6 6 6 6 6 6 2 2 2 2 2 2 2 3 3 3 3 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 0 0 -1 2 2 2 -2 -2 -2 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 0 0 -1 -2 -2 2 -2 -2 2 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 0 0 -1 -2 -2 2 2 2 -2 0 0 0 0 1 -1 1 1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ20 2 2 2 2 0 0 0 0 0 0 -1 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ21 2 2 -2 -2 0 0 0 0 0 0 -1 2 -2 -2 -2 2 2 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 0 0 -1 -2 2 -2 -2 2 -2 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 0 0 -1 -2 2 -2 2 -2 2 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 0 0 -1 2 -2 -2 2 -2 -2 0 0 0 0 -1 -1 -1 1 1 1 -1 -1 1 orthogonal lifted from D6 ρ25 2 -2 2 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 -2i 2i -2i 2i 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 -2 2 0 0 0 0 0 0 2 0 0 0 0 0 0 -2i -2i 2i 2i -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 2 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 2i -2i 2i -2i 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 -2 -2 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2i 2i -2i -2i -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 4 -4 4 -4 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ30 4 -4 -4 4 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2

Smallest permutation representation of C2×Q83S3
On 48 points
Generators in S48
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 24 3 22)(2 23 4 21)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 30 15 32)(14 29 16 31)(17 26 19 28)(18 25 20 27)(33 45 35 47)(34 48 36 46)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34)(2 33)(3 36)(4 35)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 47)(22 46)(23 45)(24 48)

G:=sub<Sym(48)| (1,34)(2,35)(3,36)(4,33)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,24,3,22)(2,23,4,21)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,30,15,32)(14,29,16,31)(17,26,19,28)(18,25,20,27)(33,45,35,47)(34,48,36,46), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,34)(2,33)(3,36)(4,35)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,47)(22,46)(23,45)(24,48)>;

G:=Group( (1,34)(2,35)(3,36)(4,33)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,24,3,22)(2,23,4,21)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,30,15,32)(14,29,16,31)(17,26,19,28)(18,25,20,27)(33,45,35,47)(34,48,36,46), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,34)(2,33)(3,36)(4,35)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,47)(22,46)(23,45)(24,48) );

G=PermutationGroup([[(1,34),(2,35),(3,36),(4,33),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(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)], [(1,24,3,22),(2,23,4,21),(5,44,7,42),(6,43,8,41),(9,40,11,38),(10,39,12,37),(13,30,15,32),(14,29,16,31),(17,26,19,28),(18,25,20,27),(33,45,35,47),(34,48,36,46)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,45,10),(6,46,11),(7,47,12),(8,48,9),(21,30,25),(22,31,26),(23,32,27),(24,29,28),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(1,34),(2,33),(3,36),(4,35),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,47),(22,46),(23,45),(24,48)]])

Matrix representation of C2×Q83S3 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 12 3 0 0 8 1
,
 12 0 0 0 0 12 0 0 0 0 5 0 0 0 12 8
,
 12 1 0 0 12 0 0 0 0 0 1 0 0 0 0 1
,
 0 12 0 0 12 0 0 0 0 0 1 0 0 0 5 12
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,8,0,0,3,1],[12,0,0,0,0,12,0,0,0,0,5,12,0,0,0,8],[12,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,12,0,0,0,0,0,1,5,0,0,0,12] >;

C2×Q83S3 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C2xQ8:3S3");
// GroupNames label

G:=SmallGroup(96,213);
// by ID

G=gap.SmallGroup(96,213);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,579,159,69,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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