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## G = C3×C32⋊2D8order 432 = 24·33

### Direct product of C3 and C32⋊2D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊2D8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×D12 — C3×C32⋊2D8
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊2D8
 Upper central C1 — C6 — C12

Generators and relations for C3×C322D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 496 in 134 conjugacy classes, 36 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C62, D4⋊S3, C3×D8, S3×C32, C32×C6, C3×C3⋊C8, C324C8, C3×D12, C3×D12, D4×C32, C32×C12, S3×C3×C6, C322D8, C3×D4⋊S3, C3×C324C8, C32×D12, C3×C322D8
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, S32, S3×C6, D4⋊S3, C3×D8, D6⋊S3, C3×C3⋊D4, C3×S32, C322D8, C3×D4⋊S3, C3×D6⋊S3, C3×C322D8

Smallest permutation representation of C3×C322D8
On 48 points
Generators in S48
(1 11 46)(2 12 47)(3 13 48)(4 14 41)(5 15 42)(6 16 43)(7 9 44)(8 10 45)(17 33 31)(18 34 32)(19 35 25)(20 36 26)(21 37 27)(22 38 28)(23 39 29)(24 40 30)
(1 11 46)(2 47 12)(3 13 48)(4 41 14)(5 15 42)(6 43 16)(7 9 44)(8 45 10)(17 33 31)(18 32 34)(19 35 25)(20 26 36)(21 37 27)(22 28 38)(23 39 29)(24 30 40)
(1 46 11)(2 12 47)(3 48 13)(4 14 41)(5 42 15)(6 16 43)(7 44 9)(8 10 45)(17 33 31)(18 32 34)(19 35 25)(20 26 36)(21 37 27)(22 28 38)(23 39 29)(24 30 40)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 33)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6AA 8A 8B 12A 12B 12C ··· 12N 24A 24B 24C 24D order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 6 6 6 ··· 6 6 6 6 6 ··· 6 8 8 12 12 12 ··· 12 24 24 24 24 size 1 1 12 12 1 1 2 ··· 2 4 4 4 2 1 1 2 ··· 2 4 4 4 12 ··· 12 18 18 2 2 4 ··· 4 18 18 18 18

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + - image C1 C2 C2 C3 C6 C6 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×D8 C3×C3⋊D4 S32 D4⋊S3 D6⋊S3 C3×S32 C32⋊2D8 C3×D4⋊S3 C3×D6⋊S3 C3×C32⋊2D8 kernel C3×C32⋊2D8 C3×C32⋊4C8 C32×D12 C32⋊2D8 C32⋊4C8 C3×D12 C3×D12 C32×C6 C3×C12 C33 D12 C3×C6 C3×C6 C12 C32 C6 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 2 1 2 2 4 4 2 4 4 8 1 2 1 2 2 4 2 4

Matrix representation of C3×C322D8 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 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 0 72 0 0 0 0 1 72
,
 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 0 0 0 0 0 0 1
,
 10 59 0 0 0 0 0 22 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 10 59 0 0 0 0 54 63 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,59,22,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,54,0,0,0,0,59,63,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×C322D8 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2D_8
% in TeX

G:=Group("C3xC3^2:2D8");
// GroupNames label

G:=SmallGroup(432,418);
// by ID

G=gap.SmallGroup(432,418);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,1011,514,80,2028,14118]);
// Polycyclic

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

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