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G = S3×C2×C8order 96 = 25·3

Direct product of C2×C8 and S3

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 130 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×4], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C2×C8, C2×C8 [×5], C22×C4, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C22×C8, S3×C8 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, S3×C2×C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, D6 [×3], C2×C8 [×6], C22×C4, C4×S3 [×2], C22×S3, C22×C8, S3×C8 [×2], S3×C2×C4, S3×C2×C8

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 S3 D6 D6 C4×S3 C4×S3 S3×C8 kernel S3×C2×C8 S3×C8 C2×C3⋊C8 C2×C24 S3×C2×C4 C4×S3 C2×Dic3 C22×S3 D6 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 16 1 2 1 2 2 8

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

 72 0 0 0 72 0 0 0 72
,
 72 0 0 0 22 0 0 0 22
,
 1 0 0 0 72 72 0 1 0
,
 72 0 0 0 72 0 0 1 1
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[72,0,0,0,22,0,0,0,22],[1,0,0,0,72,1,0,72,0],[72,0,0,0,72,1,0,0,1] >;

S3×C2×C8 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_8
% in TeX

G:=Group("S3xC2xC8");
// GroupNames label

G:=SmallGroup(96,106);
// by ID

G=gap.SmallGroup(96,106);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,50,69,2309]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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