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## G = C2×C8⋊S3order 96 = 25·3

### Direct product of C2 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×C8⋊S3
 Lower central C3 — C6 — C2×C8⋊S3
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for C2×C8⋊S3
G = < a,b,c,d | a2=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 130 in 68 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C2×C8⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4×S3, C22×S3, C2×M4(2), C8⋊S3, S3×C2×C4, C2×C8⋊S3

Smallest permutation representation of C2×C8⋊S3
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 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 45 9)(2 46 10)(3 47 11)(4 48 12)(5 41 13)(6 42 14)(7 43 15)(8 44 16)(17 26 37)(18 27 38)(19 28 39)(20 29 40)(21 30 33)(22 31 34)(23 32 35)(24 25 36)
(2 6)(4 8)(9 45)(10 42)(11 47)(12 44)(13 41)(14 46)(15 43)(16 48)(17 21)(19 23)(25 36)(26 33)(27 38)(28 35)(29 40)(30 37)(31 34)(32 39)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 24 ··· 24 size 1 1 1 1 6 6 2 1 1 1 1 6 6 2 2 2 2 2 2 2 6 6 6 6 2 2 2 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) C4×S3 C4×S3 C8⋊S3 kernel C2×C8⋊S3 C8⋊S3 C2×C3⋊C8 C2×C24 S3×C2×C4 C4×S3 C2×Dic3 C22×S3 C2×C8 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 1 2 1 4 2 2 8

Matrix representation of C2×C8⋊S3 in GL3(𝔽73) generated by

 72 0 0 0 1 0 0 0 1
,
 72 0 0 0 65 16 0 57 8
,
 1 0 0 0 72 1 0 72 0
,
 72 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[72,0,0,0,65,57,0,16,8],[1,0,0,0,72,72,0,1,0],[72,0,0,0,0,1,0,1,0] >;

C2×C8⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes S_3
% in TeX

G:=Group("C2xC8:S3");
// GroupNames label

G:=SmallGroup(96,107);
// by ID

G=gap.SmallGroup(96,107);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,50,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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