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## G = C32⋊C4⋊C8order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C32⋊C4⋊C8
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C12.29D6 — C32⋊C4⋊C8
 Lower central C32 — C3⋊S3 — C32⋊C4⋊C8
 Upper central C1 — C4

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 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 8A ··· 8H 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 6 6 8 ··· 8 12 12 12 12 24 ··· 24 size 1 1 9 9 4 4 1 1 9 9 18 18 18 18 4 4 6 ··· 6 4 4 4 4 12 ··· 12

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 4 4 type + + + - + + image C1 C2 C2 C4 C8 Q8 D4 M4(2) S3≀C2 C3⋊S3.Q8 C32⋊C4⋊C8 kernel C32⋊C4⋊C8 C12.29D6 C4×C32⋊C4 C2×C32⋊C4 C32⋊C4 C3⋊Dic3 C3×C12 C3⋊S3 C4 C2 C1 # reps 1 2 1 4 8 1 1 2 4 4 8

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 21 0 0 0 0 53 45 0 0 0 0 0 0 0 0 0 46 0 0 0 0 46 0 0 0 27 0 0 0 0 0 0 27 0 0
,
 0 48 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 72 0 0 0 0 0 0 72 0 0

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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