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G = C2xC4xC32:C4  order 288 = 25·32

Direct product of C2xC4 and C32:C4

direct product, metabelian, soluble, monomial, A-group

Aliases: C2xC4xC32:C4, (C6xC12):3C4, C3:S3:2C42, (C3xC6):1C42, C32:1(C2xC42), C62.16(C2xC4), (C4xC3:S3):8C4, (C3xC12):2(C2xC4), (C2xC3:Dic3):12C4, C3:Dic3:16(C2xC4), C3:S3.7(C22xC4), C2.2(C22xC32:C4), (C2xC3:S3).34C23, (C4xC3:S3).98C22, (C3xC6).27(C22xC4), C22.17(C2xC32:C4), (C2xC32:C4).28C22, (C22xC32:C4).11C2, (C22xC3:S3).95C22, (C2xC4xC3:S3).28C2, (C2xC3:S3).47(C2xC4), SmallGroup(288,932)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2xC4xC32:C4
Chief series
C1 — C32 — C3:S3 — C2xC3:S3 — C2xC32:C4 — C22xC32:C4 — C2xC4xC32:C4
Lower central
C32 — C2xC4xC32:C4
Upper central
C1 — C2xC4

Generators and relations for C2xC4xC32: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, C2, C3, C4, C4, C22, C22, S3, C6, C2xC4, C2xC4, C23, C32, Dic3, C12, D6, C2xC6, C42, C22xC4, C3:S3, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C22xS3, C2xC42, C3:Dic3, C3xC12, C32:C4, C2xC3:S3, C2xC3:S3, C62, S3xC2xC4, C4xC3:S3, C2xC3:Dic3, C6xC12, C2xC32:C4, C22xC3:S3, C4xC32:C4, C2xC4xC3:S3, C22xC32:C4, C2xC4xC32:C4
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, C22xC4, C2xC42, C32:C4, C2xC32:C4, C4xC32:C4, C22xC32:C4, C2xC4xC32:C4

Smallest permutation representation of C2xC4xC32:C4
►On 48 points
Generators in S48
(1 11)(2 12)(3 9)(4 10)(5 26)(6 27)(7 28)(8 25)(13 39)(14 40)(15 37)(16 38)(17 23)(18 24)(19 21)(20 22)(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 37 35)(30 38 36)(31 39 33)(32 40 34)

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E···4X6A···6F12A···12H
order122222223344444···46···612···12
size111199994411119···94···44···4

48 irreducible representations

dim111111114444
type+++++++
imageC1C2C2C2C4C4C4C4C32:C4C2xC32:C4C2xC32:C4C4xC32:C4
kernelC2xC4xC32:C4C4xC32:C4C2xC4xC3:S3C22xC32:C4C4xC3:S3C2xC3:Dic3C6xC12C2xC32:C4C2xC4C4C22C2
# reps1412422162428

Matrix representation of C2xC4xC32:C4 ►in GL5(F13)

10000
012000
001200
000120
000012
,
80000
012000
001200
000120
000012
,
10000
01000
00100
000012
000112
,
10000
012100
012000
000012
000112
,
80000
000120
000012
00100
01000

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] >;
 

C2xC4xC32: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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