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## G = C3×S3×C3⋊C8order 432 = 24·33

### Direct product of C3, S3 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×S3×C3⋊C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — S3×C3×C12 — C3×S3×C3⋊C8
 Lower central C32 — C3×S3×C3⋊C8
 Upper central C1 — C12

Generators and relations for C3×S3×C3⋊C8
G = < a,b,c,d,e | a3=b3=c2=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 304 in 126 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, S3×C8, C2×C3⋊C8, C2×C24, S3×C32, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C324C8, C3×C24, S3×C12, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, S3×C3⋊C8, S3×C24, C6×C3⋊C8, C32×C3⋊C8, C3×C324C8, S3×C3×C12, C3×S3×C3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S32, S3×C6, S3×C8, C2×C3⋊C8, C2×C24, C3×C3⋊C8, S3×Dic3, S3×C12, C6×Dic3, C3×S32, S3×C3⋊C8, S3×C24, C6×C3⋊C8, C3×S3×Dic3, C3×S3×C3⋊C8

Smallest permutation representation of C3×S3×C3⋊C8
On 48 points
Generators in S48
(1 30 35)(2 31 36)(3 32 37)(4 25 38)(5 26 39)(6 27 40)(7 28 33)(8 29 34)(9 41 20)(10 42 21)(11 43 22)(12 44 23)(13 45 24)(14 46 17)(15 47 18)(16 48 19)
(1 30 35)(2 31 36)(3 32 37)(4 25 38)(5 26 39)(6 27 40)(7 28 33)(8 29 34)(9 20 41)(10 21 42)(11 22 43)(12 23 44)(13 24 45)(14 17 46)(15 18 47)(16 19 48)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(1 35 30)(2 31 36)(3 37 32)(4 25 38)(5 39 26)(6 27 40)(7 33 28)(8 29 34)(9 41 20)(10 21 42)(11 43 22)(12 23 44)(13 45 24)(14 17 46)(15 47 18)(16 19 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)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 6N 6O 6P ··· 6U 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12P 12Q 12R 12S 12T 12U ··· 12Z 12AA ··· 12AF 24A ··· 24H 24I ··· 24T 24U ··· 24AB order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 6 ··· 6 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 3 3 1 1 2 ··· 2 4 4 4 1 1 3 3 1 1 2 ··· 2 3 3 3 3 4 4 4 6 ··· 6 3 3 3 3 9 9 9 9 1 1 1 1 2 ··· 2 3 3 3 3 4 ··· 4 6 ··· 6 3 ··· 3 6 ··· 6 9 ··· 9

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 S3 S3 Dic3 D6 Dic3 C3×S3 C3×S3 C3⋊C8 C4×S3 C3×Dic3 S3×C6 C3×Dic3 S3×C8 C3×C3⋊C8 S3×C12 S3×C24 S32 S3×Dic3 C3×S32 S3×C3⋊C8 C3×S3×Dic3 C3×S3×C3⋊C8 kernel C3×S3×C3⋊C8 C32×C3⋊C8 C3×C32⋊4C8 S3×C3×C12 S3×C3⋊C8 C32×Dic3 S3×C3×C6 C3×C3⋊C8 C32⋊4C8 S3×C12 S3×C32 C3×Dic3 S3×C6 C3×S3 C3×C3⋊C8 S3×C12 C3×Dic3 C3×C12 S3×C6 C3⋊C8 C4×S3 C3×S3 C3×C6 Dic3 C12 D6 C32 S3 C6 C3 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 8 4 4 16 1 1 1 2 1 2 2 4 2 2 4 2 4 8 4 8 1 1 2 2 2 4

Matrix representation of C3×S3×C3⋊C8 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 72 72 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 72 1 0 0 72 0
,
 51 0 0 0 0 51 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[51,0,0,0,0,51,0,0,0,0,0,1,0,0,1,0] >;

C3×S3×C3⋊C8 in GAP, Magma, Sage, TeX

C_3\times S_3\times C_3\rtimes C_8
% in TeX

G:=Group("C3xS3xC3:C8");
// GroupNames label

G:=SmallGroup(432,414);
// by ID

G=gap.SmallGroup(432,414);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,92,80,2028,14118]);
// Polycyclic

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

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