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G = C2xC32:C18order 324 = 22·34

Direct product of C2 and C32:C18

direct product, metabelian, supersoluble, monomial

Aliases: C2xC32:C18, C3:S3:C18, (C3xC6):C18, (C3xC9):5D6, (C3xC18):1S3, C6.5(S3xC9), C32:(C2xC18), C3.2(S3xC18), C32:C9:2C22, (C32xC6).3C6, C33.1(C2xC6), C32.14(S3xC6), C6.14(C32:C6), (C2xC3:S3):C9, (C3xC3:S3).C6, (C6xC3:S3).C3, (C2xC32:C9):1C2, (C3xC6).30(C3xS3), C3.5(C2xC32:C6), SmallGroup(324,62)

Series: Derived Chief Lower central Upper central

C1C32 — C2xC32:C18
C1C3C32C33C32:C9C32:C18 — C2xC32:C18
C32 — C2xC32:C18
C1C6

Generators and relations for C2xC32:C18
 G = < a,b,c,d | a2=b3=c3=d18=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 253 in 67 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2xC6, C18, C3xS3, C3:S3, C3xC6, C3xC6, C3xC9, C3xC9, C33, C2xC18, S3xC6, C2xC3:S3, S3xC9, C3xC18, C3xC18, C3xC3:S3, C32xC6, C32:C9, S3xC18, C6xC3:S3, C32:C18, C2xC32:C9, C2xC32:C18
Quotients: C1, C2, C3, C22, S3, C6, C9, D6, C2xC6, C18, C3xS3, C2xC18, S3xC6, S3xC9, C32:C6, S3xC18, C2xC32:C6, C32:C18, C2xC32:C18

Smallest permutation representation of C2xC32:C18
On 36 points
Generators in S36
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)
(2 8 14)(3 9 15)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(22 28 34)(23 29 35)
(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 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)

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

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

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

60 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F3G3H6A6B6C6D6E6F6G6H6I6J6K6L9A···9F9G···9L18A···18F18G···18L18M···18X
order1222333333336666666666669···99···918···1818···1818···18
size1199112226661122266699993···36···63···36···69···9

60 irreducible representations

dim1111111112222226666
type+++++++
imageC1C2C2C3C6C6C9C18C18S3D6C3xS3S3xC6S3xC9S3xC18C32:C6C2xC32:C6C32:C18C2xC32:C18
kernelC2xC32:C18C32:C18C2xC32:C9C6xC3:S3C3xC3:S3C32xC6C2xC3:S3C3:S3C3xC6C3xC18C3xC9C3xC6C32C6C3C6C3C2C1
# reps12124261261122661122

Matrix representation of C2xC32:C18 in GL6(F19)

1800000
0180000
0018000
0001800
0000180
0000018
,
1187039
070000
0011000
000100
0000110
000007
,
1100111413
0110000
0011000
000700
000070
000007
,
4121616011
0000012
0001800
0120000
0012000
91171315

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,18,7,0,0,0,0,7,0,11,0,0,0,0,0,0,1,0,0,3,0,0,0,11,0,9,0,0,0,0,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,11,0,0,7,0,0,14,0,0,0,7,0,13,0,0,0,0,7],[4,0,0,0,0,9,12,0,0,12,0,11,16,0,0,0,12,7,16,0,18,0,0,1,0,0,0,0,0,3,11,12,0,0,0,15] >;

C2xC32:C18 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes C_{18}
% in TeX

G:=Group("C2xC3^2:C18");
// GroupNames label

G:=SmallGroup(324,62);
// by ID

G=gap.SmallGroup(324,62);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,2164,1096,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^3=d^18=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
// generators/relations

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