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G = C8×C32⋊C4  order 288 = 25·32

Direct product of C8 and C32⋊C4

direct product, metabelian, soluble, monomial, A-group

Aliases: C8×C32⋊C4, (C3×C24)⋊1C4, C32⋊2(C4×C8), C32⋊2C8⋊7C4, C32⋊4C8⋊9C4, (C3×C6).1C42, (C8×C3⋊S3).8C2, C3⋊S3.5(C2×C8), C2.1(C4×C32⋊C4), (C2×C32⋊C4).8C4, C4.17(C2×C32⋊C4), (C3×C12).10(C2×C4), (C4×C32⋊C4).11C2, C3⋊S3⋊3C8.10C2, (C4×C3⋊S3).78C22, C3⋊Dic3.25(C2×C4), (C2×C3⋊S3).22(C2×C4), SmallGroup(288,414)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C8×C32⋊C4
Chief series
C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C4×C32⋊C4 — C8×C32⋊C4
Lower central
C32 — C8×C32⋊C4
Upper central
C1 — C8

Generators and relations for C8×C32⋊C4
 G = < a,b,c,d | a8=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Subgroups: 288 in 66 conjugacy classes, 26 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, C42, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C4×C8, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, S3×C8, C32⋊4C8, C3×C24, C32⋊2C8, C4×C3⋊S3, C2×C32⋊C4, C8×C3⋊S3, C3⋊S3⋊3C8, C4×C32⋊C4, C8×C32⋊C4
Quotients: C1, C2, C4, C22, C8, C2×C4, C42, C2×C8, C4×C8, C32⋊C4, C2×C32⋊C4, C4×C32⋊C4, C8×C32⋊C4

Smallest permutation representation of C8×C32⋊C4
►On 48 points
Generators in S48
(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 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)

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

48 conjugacy classes

class 1 2A2B2C3A3B4A4B4C···4L6A6B8A8B8C8D8E···8P12A12B12C12D24A···24H
order122233444···46688888···81212121224···24
size119944119···94411119···944444···4

48 irreducible representations

dim1111111114444
type++++++
imageC1C2C2C2C4C4C4C4C8C32⋊C4C2×C32⋊C4C4×C32⋊C4C8×C32⋊C4
kernelC8×C32⋊C4C8×C3⋊S3C3⋊S3⋊3C8C4×C32⋊C4C32⋊4C8C3×C24C32⋊2C8C2×C32⋊C4C32⋊C4C8C4C2C1
# reps11112244162248

Matrix representation of C8×C32⋊C4 ►in GL4(𝔽73) generated by

22000
02200
00220
00022
,
07200
17200
00072
00172
,
1000
0100
00721
00720
,
00720
00072
17200
07200
G:=sub<GL(4,GF(73))| [22,0,0,0,0,22,0,0,0,0,22,0,0,0,0,22],[0,1,0,0,72,72,0,0,0,0,0,1,0,0,72,72],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[0,0,1,0,0,0,72,72,72,0,0,0,0,72,0,0] >;
 

C8×C32⋊C4 in GAP, Magma, Sage, TeX 

C_8\times C_3^2\rtimes C_4
 
% in TeX
 
G:=Group("C8xC3^2:C4");
 
// GroupNames label
 
G:=SmallGroup(288,414);
 
// by ID
 
G=gap.SmallGroup(288,414);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,80,9413,691,12550,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^3=c^3=d^4=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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