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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 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C8, C2×C8, M4(2) [×4], C22×C4, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C2×M4(2), C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C2×C8⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, C4×S3 [×2], C22×S3, C2×M4(2), C8⋊S3 [×2], S3×C2×C4, C2×C8⋊S3

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

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

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

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

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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