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## G = C2×C4⋊(C32⋊C4)  order 288 = 25·32

### Direct product of C2 and C4⋊(C32⋊C4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — 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 — C3×C6 — C2×C4⋊(C32⋊C4)
 Upper central C1 — C22 — 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, ebe-1=b-1, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 736 in 146 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C22×C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C4⋊C4, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C2×C32⋊C4, C2×C32⋊C4, C22×C3⋊S3, C4⋊(C32⋊C4), C2×C4×C3⋊S3, C22×C32⋊C4, C2×C4⋊(C32⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C32⋊C4, C2×C32⋊C4, C4⋊(C32⋊C4), 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 28)(10 25)(11 26)(12 27)(13 41)(14 42)(15 43)(16 44)(29 33)(30 34)(31 35)(32 36)(37 45)(38 46)(39 47)(40 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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 30 47)(14 31 48)(15 32 45)(16 29 46)(17 21 28)(18 22 25)(19 23 26)(20 24 27)(33 38 44)(34 39 41)(35 40 42)(36 37 43)
(13 47 30)(14 48 31)(15 45 32)(16 46 29)(33 44 38)(34 41 39)(35 42 40)(36 43 37)
(1 34 17 30)(2 33 18 29)(3 36 19 32)(4 35 20 31)(5 41 28 47)(6 44 25 46)(7 43 26 45)(8 42 27 48)(9 39 21 13)(10 38 22 16)(11 37 23 15)(12 40 24 14)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + - + + + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 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×C3⋊S3 C2×C3⋊S3 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 2 2 2 2 4 2 8

Matrix representation of C2×C4⋊(C32⋊C4) in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 6 0 0 0 0 9 6 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 6 7 0 0 0 0 8 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 12 12 0 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,9,0,0,0,0,6,6,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[6,8,0,0,0,0,7,7,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C4⋊(C32⋊C4) in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4:(C3^2:C4)");
// GroupNames label

G:=SmallGroup(288,933);
// by ID

G=gap.SmallGroup(288,933);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,422,100,9413,362,12550,1203]);
// 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,e*b*e^-1=b^-1,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
// generators/relations

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