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G = C32⋊5D18  order 324 = 22·34

2nd semidirect product of C32 and D18 acting via D18/C9=C22

metabelian, supersoluble, monomial, A-group

Aliases: C32⋊5D18, C33.10D6, C9⋊2S32, C9⋊S3⋊5S3, C3⋊S3⋊2D9, C3⋊3(S3×D9), (C3×C9)⋊14D6, C32.9S32, (C32×C9)⋊5C22, C3.1(C32⋊4D6), (C9×C3⋊S3)⋊3C2, (C3×C9⋊S3)⋊3C2, (C3×C3⋊S3).5S3, SmallGroup(324,123)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C9 — C32⋊5D18
Chief series
C1 — C3 — C32 — C3×C9 — C32×C9 — C9×C3⋊S3 — C32⋊5D18
Lower central
C32×C9 — C32⋊5D18
Upper central
C1

Generators and relations for C32⋊5D18
 G = < a,b,c,d | a3=b3=c18=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 628 in 86 conjugacy classes, 20 normal (10 characteristic)
C1, C2, C3, C3, C3, C22, S3, C6, C9, C9, C32, C32, C32, D6, D9, C18, C3×S3, C3⋊S3, C3⋊S3, C3×C9, C3×C9, C33, D18, S32, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C3×C3⋊S3, C32×C9, S3×D9, C32⋊4D6, C3×C9⋊S3, C9×C3⋊S3, C32⋊5D18
Quotients: C1, C2, C22, S3, D6, D9, D18, S32, S3×D9, C32⋊4D6, C32⋊5D18

Smallest permutation representation of C32⋊5D18
►On 36 points
Generators in S36
(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 31 25)(20 26 32)(21 33 27)(22 28 34)(23 35 29)(24 30 36)

(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 25 31)(20 32 26)(21 27 33)(22 34 28)(23 29 35)(24 36 30)

(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)

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

33 conjugacy classes

class 1 2A2B2C3A3B3C3D···3H6A6B6C9A9B9C9D···9O18A18B18C
order12223333···36669999···9181818
size1927272224···41854542224···4181818

33 irreducible representations

dim11122222244444
type++++++++++++
imageC1C2C2S3S3D6D6D9D18S32S32S3×D9C32⋊4D6C32⋊5D18
kernelC32⋊5D18C3×C9⋊S3C9×C3⋊S3C9⋊S3C3×C3⋊S3C3×C9C33C3⋊S3C32C9C32C3C3C1
# reps12121213312626

Matrix representation of C32⋊5D18 ►in GL6(𝔽19)

100000
010000
0001800
0011800
000010
000001
,
0180000
1180000
001000
000100
000010
000001
,
010000
100000
000100
001000
0000140
0000015
,
0180000
1800000
0018000
0001800
0000011
000070

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,14,0,0,0,0,0,0,15],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,7,0,0,0,0,11,0] >;
 

C32⋊5D18 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_5D_{18}
 
% in TeX
 
G:=Group("C3^2:5D18");
 
// GroupNames label
 
G:=SmallGroup(324,123);
 
// by ID
 
G=gap.SmallGroup(324,123);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,80,1593,453,2164,3899]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^18=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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