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G = C8×C32⋊C4order 288 = 25·32

Direct product of C8 and C32⋊C4

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

Aliases: C8×C32⋊C4, (C3×C24)⋊1C4, C322(C4×C8), C322C87C4, C324C89C4, (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⋊S33C8.10C2, (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

C1C32 — C8×C32⋊C4
C1C32C3×C6C2×C3⋊S3C4×C3⋊S3C4×C32⋊C4 — C8×C32⋊C4
C32 — C8×C32⋊C4
C1C8

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 [×2], C3 [×2], C4, C4 [×5], C22, S3 [×4], C6 [×2], C8, C8 [×3], C2×C4 [×3], C32, Dic3 [×2], C12 [×2], D6 [×2], C42, C2×C8 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], C4×S3 [×2], C4×C8, C3⋊Dic3, C3×C12, C32⋊C4 [×4], C2×C3⋊S3, S3×C8 [×2], C324C8, C3×C24, C322C8 [×2], C4×C3⋊S3, C2×C32⋊C4 [×2], C8×C3⋊S3, C3⋊S33C8, C4×C32⋊C4, C8×C32⋊C4
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], C42, C2×C8 [×2], 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 25 35)(2 26 36)(3 27 37)(4 28 38)(5 29 39)(6 30 40)(7 31 33)(8 32 34)(9 44 23)(10 45 24)(11 46 17)(12 47 18)(13 48 19)(14 41 20)(15 42 21)(16 43 22)
(9 23 44)(10 24 45)(11 17 46)(12 18 47)(13 19 48)(14 20 41)(15 21 42)(16 22 43)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 35 42 29)(18 36 43 30)(19 37 44 31)(20 38 45 32)(21 39 46 25)(22 40 47 26)(23 33 48 27)(24 34 41 28)

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

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

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

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⋊S33C8C4×C32⋊C4C324C8C3×C24C322C8C2×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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