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G = C32:3M5(2)  order 288 = 25·32

The semidirect product of C32 and M5(2) acting via M5(2)/C8=C4

metabelian, soluble, monomial

Aliases: C32:3M5(2), (C3xC24).3C4, C8.3(C32:C4), C3:Dic3.7C8, C32:2C16:3C2, C32:4C8.33C22, (C2xC3:S3).7C8, (C8xC3:S3).7C2, (C3xC6).9(C2xC8), (C4xC3:S3).11C4, (C3xC12).9(C2xC4), C4.16(C2xC32:C4), C2.3(C3:S3:3C8), SmallGroup(288,413)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — C32:3M5(2)
Chief series
C1 — C32 — C3xC6 — C3xC12 — C32:4C8 — C32:2C16 — C32:3M5(2)
Lower central
C32 — C3xC6 — C32:3M5(2)
Upper central
C1 — C4 — C8

Generators and relations for C32:3M5(2)
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=ab-1, dad=a-1, cbc-1=a-1b-1, dbd=b-1, dcd=c9 >

Subgroups: 208 in 48 conjugacy classes, 16 normal (14 characteristic)
Quotients: C1, C2, C4, C22, C8, C2xC4, C2xC8, M5(2), C32:C4, C2xC32:C4, C3:S3:3C8, C32:3M5(2)
C1
C2
18C2
2C3
2C3
C4
9C22
9C4
2C6
2C6
6S3
6S3
6S3
6S3
C32
C8
9C8
9C2xC4
2C12
2C12
6Dic3
6D6
6D6
6Dic3
C3xC6
2C3:S3
9C2xC8
9C16
9C16
2C24
2C24
6C4xS3
6C3:C8
6C3:C8
6C4xS3
C3:Dic3
C3xC12
C2xC3:S3
9M5(2)
6S3xC8
6S3xC8
C4xC3:S3
C32:4C8
C3xC24
C32:2C16
C32:2C16
C8xC3:S3
C32:3M5(2)

Smallest permutation representation of C32:3M5(2)
►On 48 points
Generators in S48
(1 24 36)(2 25 37)(3 38 26)(4 39 27)(5 28 40)(6 29 41)(7 42 30)(8 43 31)(9 32 44)(10 17 45)(11 46 18)(12 47 19)(13 20 48)(14 21 33)(15 34 22)(16 35 23)

(2 37 25)(4 27 39)(6 41 29)(8 31 43)(10 45 17)(12 19 47)(14 33 21)(16 23 35)

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

36 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B8A8B8C8D8E8F12A12B12C12D16A···16H24A···24H
order12233444668888881212121216···1624···24
size111844111844229999444418···184···4

36 irreducible representations

dim111111124444
type+++++
imageC1C2C2C4C4C8C8M5(2)C32:C4C2xC32:C4C3:S3:3C8C32:3M5(2)
kernelC32:3M5(2)C32:2C16C8xC3:S3C3xC24C4xC3:S3C3:Dic3C2xC3:S3C32C8C4C2C1
# reps121224442248

Matrix representation of C32:3M5(2) ►in GL6(F97)

100000
010000
0096100
0096000
0000961
0000960
,
100000
010000
001000
000100
0000096
0000196
,
0960000
5000000
000001
000010
0022000
0002200
,
9600000
010000
000100
001000
000001
000010

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[0,50,0,0,0,0,96,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,1,0,0,0,0,1,0,0,0],[96,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C32:3M5(2) in GAP, Magma, Sage, TeX 

C_3^2\rtimes_3M_5(2)
 
% in TeX
 
G:=Group("C3^2:3M5(2)");
 
// GroupNames label
 
G:=SmallGroup(288,413);
 
// by ID
 
G=gap.SmallGroup(288,413);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,64,58,80,9413,691,12550,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=b^-1,d*c*d=c^9>;
 
// generators/relations
 

Export

Subgroup lattice of C32:3M5(2) in TeX

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