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G = C6xC32:C6  order 324 = 22·34

Direct product of C6 and C32:C6

direct product, metabelian, supersoluble, monomial

Aliases: C6xC32:C6, C33:9D6, C32:C62, (C2xHe3):4C6, (C6xHe3):1C2, He3:5(C2xC6), (C32xC6):2C6, (C32xC6):3S3, C33:3(C2xC6), C32:3(S3xC6), C6.5(S3xC32), (C3xHe3):6C22, (C6xC3:S3):C3, C3:S3:(C3xC6), (C3xC6):(C3xC6), C3.2(S3xC3xC6), (C3xC3:S3):2C6, (C3xC6):1(C3xS3), (C2xC3:S3):C32, SmallGroup(324,138)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C6xC32:C6
Chief series
C1 — C3 — C32 — C33 — C3xHe3 — C3xC32:C6 — C6xC32:C6
Lower central
C32 — C6xC32:C6
Upper central
C1 — C6

Generators and relations for C6xC32:C6
 G = < a,b,c,d | a6=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 496 in 136 conjugacy classes, 46 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C32, C32, C32, D6, C2xC6, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, He3, He3, C33, C33, S3xC6, C2xC3:S3, C62, C32:C6, C2xHe3, C2xHe3, S3xC32, C3xC3:S3, C32xC6, C32xC6, C3xHe3, C2xC32:C6, S3xC3xC6, C6xC3:S3, C3xC32:C6, C6xHe3, C6xC32:C6
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2xC6, C3xS3, C3xC6, S3xC6, C62, C32:C6, S3xC32, C2xC32:C6, S3xC3xC6, C3xC32:C6, C6xC32:C6

Smallest permutation representation of C6xC32:C6
►On 36 points
Generators in S36
(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 19 13)(2 20 14)(3 21 15)(4 22 16)(5 23 17)(6 24 18)(7 31 30)(8 32 25)(9 33 26)(10 34 27)(11 35 28)(12 36 29)

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

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

60 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F···3K3L···3T6A6B6C6D6E6F···6K6L···6T6U···6AJ
order1222333333···33···3666666···66···66···6
size1199112223···36···6112223···36···69···9

60 irreducible representations

dim11111111122226666
type+++++++
imageC1C2C2C3C3C6C6C6C6S3D6C3xS3S3xC6C32:C6C2xC32:C6C3xC32:C6C6xC32:C6
kernelC6xC32:C6C3xC32:C6C6xHe3C2xC32:C6C6xC3:S3C32:C6C2xHe3C3xC3:S3C32xC6C32xC6C33C3xC6C32C6C3C2C1
# reps121621264211881122

Matrix representation of C6xC32:C6 ►in GL8(F7)

30000000
03000000
00400000
00040000
00004000
00000400
00000040
00000004
,
01000000
66000000
00010000
00001000
00100000
00000001
00000100
00000010
,
10000000
01000000
00200000
00020000
00002000
00000400
00000040
00000004
,
50000000
22000000
00000200
00000040
00000001
00200000
00040000
00001000

G:=sub<GL(8,GF(7))| [3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[5,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C6xC32:C6 in GAP, Magma, Sage, TeX 

C_6\times C_3^2\rtimes C_6
 
% in TeX
 
G:=Group("C6xC3^2:C6");
 
// GroupNames label
 
G:=SmallGroup(324,138);
 
// by ID
 
G=gap.SmallGroup(324,138);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2164,1096,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^3=c^3=d^6=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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