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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, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C8, C3×C6, C3×C6, C3×C6, C24, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C62, C62, C2×C24, C32×C6, C32×C6, C322C8, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C3×C62, C2×C322C8, C3×C322C8, C6×C3⋊Dic3, C6×C322C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C2×C8, C24, C2×C12, C32⋊C4, C2×C24, C322C8, C2×C32⋊C4, C3×C32⋊C4, C2×C322C8, C3×C322C8, C6×C32⋊C4, C6×C322C8

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