Copied to
clipboard

G = C2×C4×C32⋊C4  order 288 = 25·32

Direct product of C2×C4 and C32⋊C4

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

Aliases: C2×C4×C32⋊C4, (C6×C12)⋊3C4, C3⋊S3⋊2C42, (C3×C6)⋊1C42, C32⋊1(C2×C42), C62.16(C2×C4), (C4×C3⋊S3)⋊8C4, (C3×C12)⋊2(C2×C4), (C2×C3⋊Dic3)⋊12C4, C3⋊Dic3⋊16(C2×C4), C3⋊S3.7(C22×C4), C2.2(C22×C32⋊C4), (C2×C3⋊S3).34C23, (C4×C3⋊S3).98C22, (C3×C6).27(C22×C4), C22.17(C2×C32⋊C4), (C2×C32⋊C4).28C22, (C22×C32⋊C4).11C2, (C22×C3⋊S3).95C22, (C2×C4×C3⋊S3).28C2, (C2×C3⋊S3).47(C2×C4), SmallGroup(288,932)

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, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C42, C22×C4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C42, 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, C22×C3⋊S3, C4×C32⋊C4, C2×C4×C3⋊S3, C22×C32⋊C4, C2×C4×C32⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42, 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 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⋊C4C2×C32⋊C4C2×C32⋊C4C4×C32⋊C4
kernelC2×C4×C32⋊C4C4×C32⋊C4C2×C4×C3⋊S3C22×C32⋊C4C4×C3⋊S3C2×C3⋊Dic3C6×C12C2×C32⋊C4C2×C4C4C22C2
# reps1412422162428

Matrix representation of C2×C4×C32⋊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] >;
 

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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁