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## G = S3×C32⋊2C8order 432 = 24·33

### Direct product of S3 and C32⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×C32⋊2C8
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — S3×C3⋊Dic3 — S3×C32⋊2C8
 Lower central C33 — S3×C32⋊2C8
 Upper central C1 — C2

Generators and relations for S3×C322C8
G = < a,b,c,d,e | a3=b2=c3=d3=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 544 in 88 conjugacy classes, 22 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C2×C8, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, C62, S3×C8, S3×C32, C32×C6, C322C8, C322C8, S3×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C335C4, S3×C3×C6, C2×C322C8, C3×C322C8, C334C8, S3×C3⋊Dic3, S3×C322C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C4×S3, C32⋊C4, S3×C8, C322C8, C2×C32⋊C4, C2×C322C8, S3×C32⋊C4, S3×C322C8

Smallest permutation representation of S3×C322C8
On 48 points
Generators in S48
(1 25 35)(2 26 36)(3 27 37)(4 28 38)(5 29 39)(6 30 40)(7 31 33)(8 32 34)(9 23 44)(10 24 45)(11 17 46)(12 18 47)(13 19 48)(14 20 41)(15 21 42)(16 22 43)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)
(1 25 35)(3 37 27)(5 29 39)(7 33 31)(9 23 44)(11 46 17)(13 19 48)(15 42 21)
(1 25 35)(2 36 26)(3 37 27)(4 28 38)(5 29 39)(6 40 30)(7 33 31)(8 32 34)(9 23 44)(10 45 24)(11 46 17)(12 18 47)(13 19 48)(14 41 20)(15 42 21)(16 22 43)
(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)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 24A 24B 24C 24D order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 24 24 24 24 size 1 1 3 3 2 4 4 8 8 9 9 27 27 2 4 4 8 8 12 12 12 12 9 9 9 9 27 27 27 27 18 18 18 18 18 18

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + + + - + + - image C1 C2 C2 C2 C4 C4 C8 S3 D6 C4×S3 S3×C8 C32⋊C4 C32⋊2C8 C2×C32⋊C4 S3×C32⋊C4 S3×C32⋊2C8 kernel S3×C32⋊2C8 C3×C32⋊2C8 C33⋊4C8 S3×C3⋊Dic3 C33⋊5C4 S3×C3×C6 S3×C32 C32⋊2C8 C3⋊Dic3 C3×C6 C32 D6 S3 C6 C2 C1 # reps 1 1 1 1 2 2 8 1 1 2 4 2 4 2 2 2

Matrix representation of S3×C322C8 in GL6(𝔽73)

 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 1 0 0 0 0 72 0 0 0 0 1 72 0 0 0 72 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 51 0 0 0 51 0 0 0 0 0 0 0 0 51 0 0 0 51 0 0

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,72,0,0,0,0,1,0,0,0,0,72,72,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,51,0,0,0,0,0,0,0,51,0,0,51,0,0,0,0,0,0,0,51,0] >;

S3×C322C8 in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes_2C_8
% in TeX

G:=Group("S3xC3^2:2C8");
// GroupNames label

G:=SmallGroup(432,570);
// by ID

G=gap.SmallGroup(432,570);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,36,58,1411,298,1356,1027,14118]);
// Polycyclic

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

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