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G = C32⋊C4⋊C8order 288 = 25·32

2nd semidirect product of C32⋊C4 and C8 acting via C8/C4=C2

non-abelian, soluble, monomial

Aliases: C32⋊C42C8, C4.18S3≀C2, C322(C4⋊C8), (C3×C12).18D4, C3⋊Dic3.1Q8, C3⋊S3.3M4(2), C12.29D6.3C2, C3⋊S3.3(C2×C8), (C3×C6).1(C4⋊C4), (C2×C32⋊C4).1C4, (C4×C32⋊C4).3C2, C2.1(C3⋊S3.Q8), (C4×C3⋊S3).52C22, (C2×C3⋊S3).8(C2×C4), SmallGroup(288,380)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C32⋊C4⋊C8
C1C32C3×C6C2×C3⋊S3C4×C3⋊S3C12.29D6 — C32⋊C4⋊C8
C32C3⋊S3 — C32⋊C4⋊C8
C1C4

Generators and relations for C32⋊C4⋊C8
 G = < a,b,c,d | a3=b3=c4=d8=1, cbc-1=ab=ba, cac-1=a-1b, ad=da, dbd-1=a-1b-1, dcd-1=c-1 >

Subgroups: 272 in 60 conjugacy classes, 19 normal (15 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×4], C22, S3 [×4], C6 [×2], C8 [×2], 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 [×2], C32⋊C4, C2×C3⋊S3, S3×C8 [×2], C3×C3⋊C8 [×2], C4×C3⋊S3, C2×C32⋊C4 [×2], C12.29D6 [×2], C4×C32⋊C4, C32⋊C4⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4, Q8, C4⋊C4, C2×C8, M4(2), C4⋊C8, S3≀C2, C3⋊S3.Q8, C32⋊C4⋊C8

Smallest permutation representation of C32⋊C4⋊C8
On 48 points
Generators in S48
(1 38 31)(2 39 32)(3 40 25)(4 33 26)(5 34 27)(6 35 28)(7 36 29)(8 37 30)(9 23 48)(10 24 41)(11 17 42)(12 18 43)(13 19 44)(14 20 45)(15 21 46)(16 22 47)
(1 31 38)(3 25 40)(5 27 34)(7 29 36)(9 48 23)(11 42 17)(13 44 19)(15 46 21)
(1 12 5 16)(2 9 6 13)(3 14 7 10)(4 11 8 15)(17 37 46 26)(18 27 47 38)(19 39 48 28)(20 29 41 40)(21 33 42 30)(22 31 43 34)(23 35 44 32)(24 25 45 36)
(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)| (1,38,31)(2,39,32)(3,40,25)(4,33,26)(5,34,27)(6,35,28)(7,36,29)(8,37,30)(9,23,48)(10,24,41)(11,17,42)(12,18,43)(13,19,44)(14,20,45)(15,21,46)(16,22,47), (1,31,38)(3,25,40)(5,27,34)(7,29,36)(9,48,23)(11,42,17)(13,44,19)(15,46,21), (1,12,5,16)(2,9,6,13)(3,14,7,10)(4,11,8,15)(17,37,46,26)(18,27,47,38)(19,39,48,28)(20,29,41,40)(21,33,42,30)(22,31,43,34)(23,35,44,32)(24,25,45,36), (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( (1,38,31)(2,39,32)(3,40,25)(4,33,26)(5,34,27)(6,35,28)(7,36,29)(8,37,30)(9,23,48)(10,24,41)(11,17,42)(12,18,43)(13,19,44)(14,20,45)(15,21,46)(16,22,47), (1,31,38)(3,25,40)(5,27,34)(7,29,36)(9,48,23)(11,42,17)(13,44,19)(15,46,21), (1,12,5,16)(2,9,6,13)(3,14,7,10)(4,11,8,15)(17,37,46,26)(18,27,47,38)(19,39,48,28)(20,29,41,40)(21,33,42,30)(22,31,43,34)(23,35,44,32)(24,25,45,36), (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([(1,38,31),(2,39,32),(3,40,25),(4,33,26),(5,34,27),(6,35,28),(7,36,29),(8,37,30),(9,23,48),(10,24,41),(11,17,42),(12,18,43),(13,19,44),(14,20,45),(15,21,46),(16,22,47)], [(1,31,38),(3,25,40),(5,27,34),(7,29,36),(9,48,23),(11,42,17),(13,44,19),(15,46,21)], [(1,12,5,16),(2,9,6,13),(3,14,7,10),(4,11,8,15),(17,37,46,26),(18,27,47,38),(19,39,48,28),(20,29,41,40),(21,33,42,30),(22,31,43,34),(23,35,44,32),(24,25,45,36)], [(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)])

36 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H6A6B8A···8H12A12B12C12D24A···24H
order12223344444444668···81212121224···24
size119944119918181818446···6444412···12

36 irreducible representations

dim11111222444
type+++-++
imageC1C2C2C4C8Q8D4M4(2)S3≀C2C3⋊S3.Q8C32⋊C4⋊C8
kernelC32⋊C4⋊C8C12.29D6C4×C32⋊C4C2×C32⋊C4C32⋊C4C3⋊Dic3C3×C12C3⋊S3C4C2C1
# reps12148112448

Matrix representation of C32⋊C4⋊C8 in GL6(𝔽73)

100000
010000
0007200
0017200
0000072
0000172
,
100000
010000
0072100
0072000
000010
000001
,
28210000
53450000
0000046
0000460
0027000
0002700
,
0480000
400000
000010
000001
0072000
0007200

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,53,0,0,0,0,21,45,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,46,0,0,0,0,46,0,0,0],[0,4,0,0,0,0,48,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;

C32⋊C4⋊C8 in GAP, Magma, Sage, TeX

C_3^2\rtimes C_4\rtimes C_8
% in TeX

G:=Group("C3^2:C4:C8");
// GroupNames label

G:=SmallGroup(288,380);
// by ID

G=gap.SmallGroup(288,380);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,64,80,2693,2028,691,797,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^4=d^8=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b,a*d=d*a,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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