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G = C2×C4⋊(C32⋊C4)  order 288 = 25·32

Direct product of C2 and C4⋊(C32⋊C4)

direct product, metabelian, soluble, monomial

Aliases: C2×C4⋊(C32⋊C4), (C6×C12)⋊4C4, C62.17(C2×C4), (C4×C3⋊S3)⋊9C4, C4⋊2(C2×C32⋊C4), C32⋊3(C2×C4⋊C4), C3⋊S3⋊5(C4⋊C4), (C3×C6)⋊3(C4⋊C4), (C3×C12)⋊3(C2×C4), C3⋊S3.5(C2×Q8), (C2×C4)⋊3(C32⋊C4), C3⋊S3.10(C2×D4), (C2×C3⋊S3).49D4, (C2×C3⋊S3).10Q8, C3⋊Dic3⋊17(C2×C4), (C2×C3⋊Dic3)⋊13C4, C2.6(C22×C32⋊C4), (C2×C3⋊S3).35C23, (C4×C3⋊S3).99C22, (C3×C6).28(C22×C4), (C22×C32⋊C4).8C2, C22.18(C2×C32⋊C4), (C2×C32⋊C4).22C22, (C22×C3⋊S3).96C22, (C2×C4×C3⋊S3).29C2, (C2×C3⋊S3).48(C2×C4), SmallGroup(288,933)

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 2A2B2C2D2E2F2G3A3B4A4B4C···4L6A···6F12A···12H
order1222222233444···46···612···12
size11119999442218···184···44···4

36 irreducible representations

dim1111111224444
type+++++-+++
imageC1C2C2C2C4C4C4D4Q8C32⋊C4C2×C32⋊C4C2×C32⋊C4C4⋊(C32⋊C4)
kernelC2×C4⋊(C32⋊C4)C4⋊(C32⋊C4)C2×C4×C3⋊S3C22×C32⋊C4C4×C3⋊S3C2×C3⋊Dic3C6×C12C2×C3⋊S3C2×C3⋊S3C2×C4C4C22C2
# reps1412422222428

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

1200000
0120000
001000
000100
000010
000001
,
760000
960000
008000
000800
000050
000005
,
100000
010000
000100
00121200
000001
00001212
,
100000
010000
001000
000100
00001212
000010
,
670000
870000
000010
000001
001000
00121200

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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