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G = Dic3×C3.A4order 432 = 24·33

Direct product of Dic3 and C3.A4

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

Aliases: Dic3×C3.A4, C62.6C12, (C2×C6)⋊C36, C6.18(S3×A4), (C22×C6).C18, (C2×C62).5C6, C32.2(C4×A4), C23.2(S3×C9), (C22×Dic3)⋊C9, C3.4(Dic3×A4), (C3×Dic3).2A4, C222(C9×Dic3), C3⋊(C4×C3.A4), (Dic3×C2×C6).C3, (C3×C3.A4)⋊1C4, C2.1(S3×C3.A4), C6.3(C2×C3.A4), (C3×C6).13(C2×A4), (C2×C3.A4).3S3, (C6×C3.A4).1C2, (C2×C6).8(C3×Dic3), (C22×C6).18(C3×S3), SmallGroup(432,271)

Series: Derived Chief Lower central Upper central

C1C2×C6 — Dic3×C3.A4
C1C3C2×C6C62C2×C62C6×C3.A4 — Dic3×C3.A4
C2×C6 — Dic3×C3.A4
C1C6

Generators and relations for Dic3×C3.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 [×2], C3 [×2], C3, C4 [×2], C22, C22 [×2], C6 [×2], C6 [×7], C2×C4 [×2], C23, C9 [×2], C32, Dic3, Dic3, C12 [×2], C2×C6 [×2], C2×C6 [×7], C22×C4, C18 [×2], C3×C6, C3×C6 [×2], C2×Dic3 [×2], C2×C12 [×2], C22×C6 [×2], C22×C6, C3×C9, C36, C3.A4, C3.A4, C3×Dic3, C3×Dic3, C62, C62 [×2], C22×Dic3, C22×C12, C3×C18, C2×C3.A4, C2×C3.A4, C6×Dic3 [×2], C2×C62, C9×Dic3, C3×C3.A4, C4×C3.A4, Dic3×C2×C6, C6×C3.A4, Dic3×C3.A4
Quotients: C1, C2, C3, C4, S3, C6, C9, Dic3, C12, A4, C18, C3×S3, C2×A4, C36, C3.A4, C3×Dic3, C4×A4, S3×C9, C2×C3.A4, S3×A4, C9×Dic3, C4×C3.A4, Dic3×A4, S3×C3.A4, Dic3×C3.A4

Smallest permutation representation of Dic3×C3.A4
On 36 points
Generators in S36
(1 24 4 27 7 21)(2 25 5 19 8 22)(3 26 6 20 9 23)(10 34 16 31 13 28)(11 35 17 32 14 29)(12 36 18 33 15 30)
(1 10 27 31)(2 11 19 32)(3 12 20 33)(4 13 21 34)(5 14 22 35)(6 15 23 36)(7 16 24 28)(8 17 25 29)(9 18 26 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 27)(2 19)(4 21)(5 22)(7 24)(8 25)(10 31)(11 32)(13 34)(14 35)(16 28)(17 29)
(2 19)(3 20)(5 22)(6 23)(8 25)(9 26)(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)| (1,24,4,27,7,21)(2,25,5,19,8,22)(3,26,6,20,9,23)(10,34,16,31,13,28)(11,35,17,32,14,29)(12,36,18,33,15,30), (1,10,27,31)(2,11,19,32)(3,12,20,33)(4,13,21,34)(5,14,22,35)(6,15,23,36)(7,16,24,28)(8,17,25,29)(9,18,26,30), (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), (1,27)(2,19)(4,21)(5,22)(7,24)(8,25)(10,31)(11,32)(13,34)(14,35)(16,28)(17,29), (2,19)(3,20)(5,22)(6,23)(8,25)(9,26)(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( (1,24,4,27,7,21)(2,25,5,19,8,22)(3,26,6,20,9,23)(10,34,16,31,13,28)(11,35,17,32,14,29)(12,36,18,33,15,30), (1,10,27,31)(2,11,19,32)(3,12,20,33)(4,13,21,34)(5,14,22,35)(6,15,23,36)(7,16,24,28)(8,17,25,29)(9,18,26,30), (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), (1,27)(2,19)(4,21)(5,22)(7,24)(8,25)(10,31)(11,32)(13,34)(14,35)(16,28)(17,29), (2,19)(3,20)(5,22)(6,23)(8,25)(9,26)(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([(1,24,4,27,7,21),(2,25,5,19,8,22),(3,26,6,20,9,23),(10,34,16,31,13,28),(11,35,17,32,14,29),(12,36,18,33,15,30)], [(1,10,27,31),(2,11,19,32),(3,12,20,33),(4,13,21,34),(5,14,22,35),(6,15,23,36),(7,16,24,28),(8,17,25,29),(9,18,26,30)], [(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)], [(1,27),(2,19),(4,21),(5,22),(7,24),(8,25),(10,31),(11,32),(13,34),(14,35),(16,28),(17,29)], [(2,19),(3,20),(5,22),(6,23),(8,25),(9,26),(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+++-+++-
imageC1C2C3C4C6C9C12C18C36S3Dic3C3×S3C3×Dic3S3×C9C9×Dic3A4C2×A4C3.A4C4×A4C2×C3.A4C4×C3.A4S3×A4Dic3×A4S3×C3.A4Dic3×C3.A4
kernelDic3×C3.A4C6×C3.A4Dic3×C2×C6C3×C3.A4C2×C62C22×Dic3C62C22×C6C2×C6C2×C3.A4C3.A4C22×C6C2×C6C23C22C3×Dic3C3×C6Dic3C32C6C3C6C3C2C1
# reps11222646121122661122241122

Matrix representation of Dic3×C3.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] >;

Dic3×C3.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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