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G = C323M5(2)  order 288 = 25·32

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

metabelian, soluble, monomial

Aliases: C323M5(2), (C3×C24).3C4, C8.3(C32⋊C4), C3⋊Dic3.7C8, C322C163C2, C324C8.33C22, (C2×C3⋊S3).7C8, (C8×C3⋊S3).7C2, (C3×C6).9(C2×C8), (C4×C3⋊S3).11C4, (C3×C12).9(C2×C4), C4.16(C2×C32⋊C4), C2.3(C3⋊S33C8), SmallGroup(288,413)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C323M5(2)
C1C32C3×C6C3×C12C324C8C322C16 — C323M5(2)
C32C3×C6 — C323M5(2)
C1C4C8

Generators and relations for C323M5(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 >

18C2
2C3
2C3
9C22
9C4
2C6
2C6
6S3
6S3
6S3
6S3
9C8
9C2×C4
2C12
2C12
6Dic3
6D6
6D6
6Dic3
2C3⋊S3
9C2×C8
9C16
9C16
2C24
2C24
6C4×S3
6C3⋊C8
6C3⋊C8
6C4×S3
9M5(2)
6S3×C8
6S3×C8

Smallest permutation representation of C323M5(2)
On 48 points
Generators in S48
(1 17 48)(2 18 33)(3 34 19)(4 35 20)(5 21 36)(6 22 37)(7 38 23)(8 39 24)(9 25 40)(10 26 41)(11 42 27)(12 43 28)(13 29 44)(14 30 45)(15 46 31)(16 47 32)
(2 33 18)(4 20 35)(6 37 22)(8 24 39)(10 41 26)(12 28 43)(14 45 30)(16 32 47)
(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 40)(18 33)(19 42)(20 35)(21 44)(22 37)(23 46)(24 39)(25 48)(26 41)(27 34)(28 43)(29 36)(30 45)(31 38)(32 47)

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim111111124444
type+++++
imageC1C2C2C4C4C8C8M5(2)C32⋊C4C2×C32⋊C4C3⋊S33C8C323M5(2)
kernelC323M5(2)C322C16C8×C3⋊S3C3×C24C4×C3⋊S3C3⋊Dic3C2×C3⋊S3C32C8C4C2C1
# reps121224442248

Matrix representation of C323M5(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] >;

C323M5(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

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Subgroup lattice of C323M5(2) in TeX

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