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## G = C2×C4×C32⋊C4order 288 = 25·32

### Direct product of C2×C4 and C32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C4×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C22×C32⋊C4 — C2×C4×C32⋊C4
 Lower central C32 — C2×C4×C32⋊C4
 Upper central C1 — C2×C4

Generators and relations for C2×C4×C32⋊C4
G = < a,b,c,d,e | a2=b4=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 736 in 162 conjugacy classes, 62 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C4 [×2], C4 [×10], C22, C22 [×6], S3 [×8], C6 [×6], C2×C4, C2×C4 [×17], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C42 [×4], C22×C4 [×3], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2×C42, C3⋊Dic3 [×2], C3×C12 [×2], C32⋊C4 [×8], C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62, S3×C2×C4 [×2], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C2×C32⋊C4 [×12], C22×C3⋊S3, C4×C32⋊C4 [×4], C2×C4×C3⋊S3, C22×C32⋊C4 [×2], C2×C4×C32⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], C2×C42, C32⋊C4, C2×C32⋊C4 [×3], C4×C32⋊C4 [×2], C22×C32⋊C4, C2×C4×C32⋊C4

Smallest permutation representation of C2×C4×C32⋊C4
On 48 points
Generators in S48
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 27)(10 28)(11 25)(12 26)(13 40)(14 37)(15 38)(16 39)(29 45)(30 46)(31 47)(32 48)(33 43)(34 44)(35 41)(36 42)
(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)
(13 43 47)(14 44 48)(15 41 45)(16 42 46)(29 38 35)(30 39 36)(31 40 33)(32 37 34)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(29 38 35)(30 39 36)(31 40 33)(32 37 34)
(1 35 17 41)(2 36 18 42)(3 33 19 43)(4 34 20 44)(5 38 27 45)(6 39 28 46)(7 40 25 47)(8 37 26 48)(9 29 21 15)(10 30 22 16)(11 31 23 13)(12 32 24 14)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E ··· 4X 6A ··· 6F 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 9 9 9 9 4 4 1 1 1 1 9 ··· 9 4 ··· 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C32⋊C4 C2×C32⋊C4 C2×C32⋊C4 C4×C32⋊C4 kernel C2×C4×C32⋊C4 C4×C32⋊C4 C2×C4×C3⋊S3 C22×C32⋊C4 C4×C3⋊S3 C2×C3⋊Dic3 C6×C12 C2×C32⋊C4 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 2 16 2 4 2 8

Matrix representation of C2×C4×C32⋊C4 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 1 12
,
 1 0 0 0 0 0 12 1 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 1 12
,
 8 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 1 0 0 0 1 0 0 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,12],[1,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,12,12],[8,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,12,0,0,0,0,0,12,0,0] >;

C2×C4×C32⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3^2\rtimes C_4
% in TeX

G:=Group("C2xC4xC3^2:C4");
// GroupNames label

G:=SmallGroup(288,932);
// by ID

G=gap.SmallGroup(288,932);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,100,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
// generators/relations

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