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G = C3xDic3xA4order 432 = 24·33

Direct product of C3, Dic3 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: C3xDic3xA4, C62:6C12, C3:(C12xA4), C6.3(C6xA4), (C3xA4):5C12, (C6xA4).9C6, C6.22(S3xA4), C32:4(C4xA4), (C6xA4).12S3, (C32xA4):6C4, (C2xC62).9C6, C23.2(S3xC32), (C22xDic3):C32, C22:2(C32xDic3), (C2xC6):(C3xC12), C2.1(C3xS3xA4), (Dic3xC2xC6):C3, (A4xC3xC6).4C2, (C22xC6).(C3xC6), (C2xA4).2(C3xS3), (C3xC6).17(C2xA4), (C2xC6):4(C3xDic3), (C22xC6).20(C3xS3), SmallGroup(432,624)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C3xDic3xA4
C1C3C2xC6C22xC6C2xC62A4xC3xC6 — C3xDic3xA4
C2xC6 — C3xDic3xA4
C1C6

Generators and relations for C3xDic3xA4
 G = < a,b,c,d,e,f | a3=b6=d2=e2=f3=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 484 in 136 conjugacy classes, 40 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2xC4, C23, C32, C32, Dic3, Dic3, C12, A4, A4, C2xC6, C2xC6, C22xC4, C3xC6, C3xC6, C2xDic3, C2xC12, C2xA4, C2xA4, C22xC6, C22xC6, C33, C3xDic3, C3xDic3, C3xC12, C3xA4, C3xA4, C3xA4, C62, C62, C4xA4, C22xDic3, C22xC12, C32xC6, C6xDic3, C6xA4, C6xA4, C6xA4, C2xC62, C32xDic3, C32xA4, Dic3xA4, C12xA4, Dic3xC2xC6, A4xC3xC6, C3xDic3xA4
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, A4, C3xS3, C3xC6, C2xA4, C3xDic3, C3xC12, C3xA4, C4xA4, S3xC32, S3xA4, C6xA4, C32xDic3, Dic3xA4, C12xA4, C3xS3xA4, C3xDic3xA4

Smallest permutation representation of C3xDic3xA4
On 36 points
Generators in S36
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(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 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(1 15 7)(2 16 8)(3 17 9)(4 18 10)(5 13 11)(6 14 12)(19 33 29)(20 34 30)(21 35 25)(22 36 26)(23 31 27)(24 32 28)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F···3K3L···3Q4A4B4C4D6A6B6C6D6E6F6G6H6I6J···6O6P···6U6V···6AA12A12B12C12D12E12F12G12H12I···12T
order1222333333···33···344446666666666···66···66···6121212121212121212···12
size1133112224···48···833991122233334···46···68···83333999912···12

72 irreducible representations

dim1111111112222223333336666
type+++-+++-
imageC1C2C3C3C4C6C6C12C12S3Dic3C3xS3C3xS3C3xDic3C3xDic3A4C2xA4C3xA4C4xA4C6xA4C12xA4S3xA4Dic3xA4C3xS3xA4C3xDic3xA4
kernelC3xDic3xA4A4xC3xC6Dic3xA4Dic3xC2xC6C32xA4C6xA4C2xC62C3xA4C62C6xA4C3xA4C2xA4C22xC6A4C2xC6C3xDic3C3xC6Dic3C32C6C3C6C3C2C1
# reps11622621241162621122241122

Matrix representation of C3xDic3xA4 in GL5(F13)

90000
09000
00900
00090
00009
,
100000
34000
00100
00010
00001
,
53000
08000
001200
000120
000012
,
10000
01000
00100
000120
000012
,
10000
01000
001200
00010
000012
,
10000
01000
00009
00400
00040

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[10,3,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,3,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,9,0,0] >;

C3xDic3xA4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3\times A_4
% in TeX

G:=Group("C3xDic3xA4");
// GroupNames label

G:=SmallGroup(432,624);
// by ID

G=gap.SmallGroup(432,624);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-2,2,-3,126,1901,768,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^6=d^2=e^2=f^3=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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