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## G = C6×C32⋊2C8order 432 = 24·33

### Direct product of C6 and C32⋊2C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C322C8
G = < a,b,c,d | a6=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Subgroups: 332 in 96 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×2], C22, C6, C6 [×2], C6 [×12], C8 [×2], C2×C4, C32, C32 [×4], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×4], C2×C8, C3×C6, C3×C6 [×2], C3×C6 [×12], C24 [×2], C2×Dic3 [×2], C2×C12, C33, C3×Dic3 [×4], C3⋊Dic3 [×2], C62, C62 [×4], C2×C24, C32×C6, C32×C6 [×2], C322C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C3×C3⋊Dic3 [×2], C3×C62, C2×C322C8, C3×C322C8 [×2], C6×C3⋊Dic3, C6×C322C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C8 [×2], C2×C4, C12 [×2], C2×C6, C2×C8, C24 [×2], C2×C12, C32⋊C4, C2×C24, C322C8 [×2], C2×C32⋊C4, C3×C32⋊C4, C2×C322C8, C3×C322C8 [×2], C6×C32⋊C4, C6×C322C8

Smallest permutation representation of C6×C322C8
On 48 points
Generators in S48
(1 33 31 15 21 47)(2 34 32 16 22 48)(3 35 25 9 23 41)(4 36 26 10 24 42)(5 37 27 11 17 43)(6 38 28 12 18 44)(7 39 29 13 19 45)(8 40 30 14 20 46)
(1 31 21)(2 22 32)(3 23 25)(4 26 24)(5 27 17)(6 18 28)(7 19 29)(8 30 20)(9 35 41)(10 42 36)(11 43 37)(12 38 44)(13 39 45)(14 46 40)(15 47 33)(16 34 48)
(2 32 22)(4 24 26)(6 28 18)(8 20 30)(10 36 42)(12 44 38)(14 40 46)(16 48 34)
(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)| (1,33,31,15,21,47)(2,34,32,16,22,48)(3,35,25,9,23,41)(4,36,26,10,24,42)(5,37,27,11,17,43)(6,38,28,12,18,44)(7,39,29,13,19,45)(8,40,30,14,20,46), (1,31,21)(2,22,32)(3,23,25)(4,26,24)(5,27,17)(6,18,28)(7,19,29)(8,30,20)(9,35,41)(10,42,36)(11,43,37)(12,38,44)(13,39,45)(14,46,40)(15,47,33)(16,34,48), (2,32,22)(4,24,26)(6,28,18)(8,20,30)(10,36,42)(12,44,38)(14,40,46)(16,48,34), (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( (1,33,31,15,21,47)(2,34,32,16,22,48)(3,35,25,9,23,41)(4,36,26,10,24,42)(5,37,27,11,17,43)(6,38,28,12,18,44)(7,39,29,13,19,45)(8,40,30,14,20,46), (1,31,21)(2,22,32)(3,23,25)(4,26,24)(5,27,17)(6,18,28)(7,19,29)(8,30,20)(9,35,41)(10,42,36)(11,43,37)(12,38,44)(13,39,45)(14,46,40)(15,47,33)(16,34,48), (2,32,22)(4,24,26)(6,28,18)(8,20,30)(10,36,42)(12,44,38)(14,40,46)(16,48,34), (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([(1,33,31,15,21,47),(2,34,32,16,22,48),(3,35,25,9,23,41),(4,36,26,10,24,42),(5,37,27,11,17,43),(6,38,28,12,18,44),(7,39,29,13,19,45),(8,40,30,14,20,46)], [(1,31,21),(2,22,32),(3,23,25),(4,26,24),(5,27,17),(6,18,28),(7,19,29),(8,30,20),(9,35,41),(10,42,36),(11,43,37),(12,38,44),(13,39,45),(14,46,40),(15,47,33),(16,34,48)], [(2,32,22),(4,24,26),(6,28,18),(8,20,30),(10,36,42),(12,44,38),(14,40,46),(16,48,34)], [(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)])

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A ··· 6F 6G ··· 6X 8A ··· 8H 12A ··· 12H 24A ··· 24P order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 4 ··· 4 9 9 9 9 1 ··· 1 4 ··· 4 9 ··· 9 9 ··· 9 9 ··· 9

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 C32⋊C4 C32⋊2C8 C2×C32⋊C4 C3×C32⋊C4 C3×C32⋊2C8 C6×C32⋊C4 kernel C6×C32⋊2C8 C3×C32⋊2C8 C6×C3⋊Dic3 C2×C32⋊2C8 C3×C3⋊Dic3 C3×C62 C32⋊2C8 C2×C3⋊Dic3 C32×C6 C3⋊Dic3 C62 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 4 2 4 8 4

Matrix representation of C6×C322C8 in GL5(𝔽73)

 65 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 8 0 0 12 0 0 64 25 0 0 0 0 8 0 0 0 0 0 64
,
 1 0 0 0 0 0 1 0 58 38 0 0 1 54 67 0 0 0 64 0 0 0 0 0 8
,
 1 0 0 0 0 0 35 47 69 38 0 6 6 38 4 0 0 7 67 6 0 66 0 47 38

G:=sub<GL(5,GF(73))| [65,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,8,0,0,0,0,0,64,0,0,0,0,25,8,0,0,12,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,58,54,64,0,0,38,67,0,8],[1,0,0,0,0,0,35,6,0,66,0,47,6,7,0,0,69,38,67,47,0,38,4,6,38] >;

C6×C322C8 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes_2C_8
% in TeX

G:=Group("C6xC3^2:2C8");
// GroupNames label

G:=SmallGroup(432,632);
// by ID

G=gap.SmallGroup(432,632);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,80,14117,362,18822,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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