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G = Dic3xC3.A4order 432 = 24·33

Direct product of Dic3 and C3.A4

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

Aliases: Dic3xC3.A4, C62.6C12, (C2xC6):C36, C6.18(S3xA4), (C22xC6).C18, (C2xC62).5C6, C32.2(C4xA4), C23.2(S3xC9), (C22xDic3):C9, C3.4(Dic3xA4), (C3xDic3).2A4, C22:2(C9xDic3), C3:(C4xC3.A4), (Dic3xC2xC6).C3, (C3xC3.A4):1C4, C2.1(S3xC3.A4), C6.3(C2xC3.A4), (C3xC6).13(C2xA4), (C2xC3.A4).3S3, (C6xC3.A4).1C2, (C2xC6).8(C3xDic3), (C22xC6).18(C3xS3), SmallGroup(432,271)

Series: Derived Chief Lower central Upper central

C1C2xC6 — Dic3xC3.A4
C1C3C2xC6C62C2xC62C6xC3.A4 — Dic3xC3.A4
C2xC6 — Dic3xC3.A4
C1C6

Generators and relations for Dic3xC3.A4
 G = < a,b,c,d,e,f | a6=c3=d2=e2=1, b2=a3, f3=c, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 244 in 76 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2xC4, C23, C9, C32, Dic3, Dic3, C12, C2xC6, C2xC6, C22xC4, C18, C3xC6, C3xC6, C2xDic3, C2xC12, C22xC6, C22xC6, C3xC9, C36, C3.A4, C3.A4, C3xDic3, C3xDic3, C62, C62, C22xDic3, C22xC12, C3xC18, C2xC3.A4, C2xC3.A4, C6xDic3, C2xC62, C9xDic3, C3xC3.A4, C4xC3.A4, Dic3xC2xC6, C6xC3.A4, Dic3xC3.A4
Quotients: C1, C2, C3, C4, S3, C6, C9, Dic3, C12, A4, C18, C3xS3, C2xA4, C36, C3.A4, C3xDic3, C4xA4, S3xC9, C2xC3.A4, S3xA4, C9xDic3, C4xC3.A4, Dic3xA4, S3xC3.A4, Dic3xC3.A4

Smallest permutation representation of Dic3xC3.A4
On 36 points
Generators in S36
(1 25 4 19 7 22)(2 26 5 20 8 23)(3 27 6 21 9 24)(10 34 16 31 13 28)(11 35 17 32 14 29)(12 36 18 33 15 30)
(1 10 19 31)(2 11 20 32)(3 12 21 33)(4 13 22 34)(5 14 23 35)(6 15 24 36)(7 16 25 28)(8 17 26 29)(9 18 27 30)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 19)(2 20)(4 22)(5 23)(7 25)(8 26)(10 31)(11 32)(13 34)(14 35)(16 28)(17 29)
(2 20)(3 21)(5 23)(6 24)(8 26)(9 27)(11 32)(12 33)(14 35)(15 36)(17 29)(18 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)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I6J···6O9A···9F9G···9L12A12B12C12D12E12F12G12H18A···18F18G···18L36A···36L
order12223333344446666666666···69···99···9121212121212121218···1818···1836···36
size11331122233991122233336···64···48···8333399994···48···812···12

72 irreducible representations

dim1111111112222223333336666
type+++-+++-
imageC1C2C3C4C6C9C12C18C36S3Dic3C3xS3C3xDic3S3xC9C9xDic3A4C2xA4C3.A4C4xA4C2xC3.A4C4xC3.A4S3xA4Dic3xA4S3xC3.A4Dic3xC3.A4
kernelDic3xC3.A4C6xC3.A4Dic3xC2xC6C3xC3.A4C2xC62C22xDic3C62C22xC6C2xC6C2xC3.A4C3.A4C22xC6C2xC6C23C22C3xDic3C3xC6Dic3C32C6C3C6C3C2C1
# reps11222646121122661122241122

Matrix representation of Dic3xC3.A4 in GL5(F37)

115000
027000
00100
00010
00001
,
319000
1734000
003600
000360
000036
,
10000
01000
002600
000260
000026
,
10000
01000
003600
000112
000036
,
10000
01000
0036012
0003625
00001
,
10000
01000
000160
0021210
0001516

G:=sub<GL(5,GF(37))| [11,0,0,0,0,5,27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,17,0,0,0,19,34,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,12,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,12,25,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,21,0,0,0,16,21,15,0,0,0,0,16] >;

Dic3xC3.A4 in GAP, Magma, Sage, TeX

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

G:=Group("Dic3xC3.A4");
// GroupNames label

G:=SmallGroup(432,271);
// by ID

G=gap.SmallGroup(432,271);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-2,2,-3,42,92,1901,768,14118]);
// Polycyclic

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