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G = C33⋊6C42  order 432 = 24·33

3rd semidirect product of C33 and C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, A-group

Aliases: C33⋊6C42, C62.83D6, C3⋊2Dic32, C3⋊Dic3⋊4Dic3, C6.19(S3×Dic3), C32⋊7(C4×Dic3), C6.20(C6.D6), (C3×C62).13C22, C22.2(C32⋊4D6), (C2×C6).59S32, (C3×C6).55(C4×S3), (C3×C3⋊Dic3)⋊7C4, (C2×C3⋊Dic3).8S3, C2.2(C33⋊9(C2×C4)), (C6×C3⋊Dic3).13C2, (C32×C6).46(C2×C4), (C3×C6).42(C2×Dic3), SmallGroup(432,460)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C33⋊6C42
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — C6×C3⋊Dic3 — C33⋊6C42
Lower central
C33 — C33⋊6C42
Upper central
C1 — C22

Generators and relations for C33⋊6C42
 G = < a,b,c,d,e | a3=b3=c3=d4=e4=1, ab=ba, ac=ca, dad-1=eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, dcd-1=c-1, ce=ec, de=ed >

Subgroups: 600 in 178 conjugacy classes, 59 normal (5 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C42, C3×C6, C3×C6, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C62, C62, C4×Dic3, C32×C6, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C3×C62, Dic32, C6×C3⋊Dic3, C33⋊6C42
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C4×S3, C2×Dic3, S32, C4×Dic3, S3×Dic3, C6.D6, C32⋊4D6, Dic32, C33⋊9(C2×C4), C33⋊6C42

Smallest permutation representation of C33⋊6C42
►On 48 points
Generators in S48
(1 39 5)(2 6 40)(3 37 7)(4 8 38)(9 36 22)(10 23 33)(11 34 24)(12 21 35)(13 27 45)(14 46 28)(15 25 47)(16 48 26)(17 43 29)(18 30 44)(19 41 31)(20 32 42)

(1 39 5)(2 40 6)(3 37 7)(4 38 8)(9 22 36)(10 23 33)(11 24 34)(12 21 35)(13 27 45)(14 28 46)(15 25 47)(16 26 48)(17 43 29)(18 44 30)(19 41 31)(20 42 32)

(1 5 39)(2 40 6)(3 7 37)(4 38 8)(9 36 22)(10 23 33)(11 34 24)(12 21 35)(13 27 45)(14 46 28)(15 25 47)(16 48 26)(17 29 43)(18 44 30)(19 31 41)(20 42 32)

(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)(37 38 39 40)(41 42 43 44)(45 46 47 48)

(1 13 29 23)(2 14 30 24)(3 15 31 21)(4 16 32 22)(5 27 43 33)(6 28 44 34)(7 25 41 35)(8 26 42 36)(9 38 48 20)(10 39 45 17)(11 40 46 18)(12 37 47 19)
 
G:=sub<Sym(48)| (1,39,5)(2,6,40)(3,37,7)(4,8,38)(9,36,22)(10,23,33)(11,34,24)(12,21,35)(13,27,45)(14,46,28)(15,25,47)(16,48,26)(17,43,29)(18,30,44)(19,41,31)(20,32,42), (1,39,5)(2,40,6)(3,37,7)(4,38,8)(9,22,36)(10,23,33)(11,24,34)(12,21,35)(13,27,45)(14,28,46)(15,25,47)(16,26,48)(17,43,29)(18,44,30)(19,41,31)(20,42,32), (1,5,39)(2,40,6)(3,7,37)(4,38,8)(9,36,22)(10,23,33)(11,34,24)(12,21,35)(13,27,45)(14,46,28)(15,25,47)(16,48,26)(17,29,43)(18,44,30)(19,31,41)(20,42,32), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,13,29,23)(2,14,30,24)(3,15,31,21)(4,16,32,22)(5,27,43,33)(6,28,44,34)(7,25,41,35)(8,26,42,36)(9,38,48,20)(10,39,45,17)(11,40,46,18)(12,37,47,19)>;
 
G:=Group( (1,39,5)(2,6,40)(3,37,7)(4,8,38)(9,36,22)(10,23,33)(11,34,24)(12,21,35)(13,27,45)(14,46,28)(15,25,47)(16,48,26)(17,43,29)(18,30,44)(19,41,31)(20,32,42), (1,39,5)(2,40,6)(3,37,7)(4,38,8)(9,22,36)(10,23,33)(11,24,34)(12,21,35)(13,27,45)(14,28,46)(15,25,47)(16,26,48)(17,43,29)(18,44,30)(19,41,31)(20,42,32), (1,5,39)(2,40,6)(3,7,37)(4,38,8)(9,36,22)(10,23,33)(11,34,24)(12,21,35)(13,27,45)(14,46,28)(15,25,47)(16,48,26)(17,29,43)(18,44,30)(19,31,41)(20,42,32), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,13,29,23)(2,14,30,24)(3,15,31,21)(4,16,32,22)(5,27,43,33)(6,28,44,34)(7,25,41,35)(8,26,42,36)(9,38,48,20)(10,39,45,17)(11,40,46,18)(12,37,47,19) );
 
G=PermutationGroup([[(1,39,5),(2,6,40),(3,37,7),(4,8,38),(9,36,22),(10,23,33),(11,34,24),(12,21,35),(13,27,45),(14,46,28),(15,25,47),(16,48,26),(17,43,29),(18,30,44),(19,41,31),(20,32,42)], [(1,39,5),(2,40,6),(3,37,7),(4,38,8),(9,22,36),(10,23,33),(11,24,34),(12,21,35),(13,27,45),(14,28,46),(15,25,47),(16,26,48),(17,43,29),(18,44,30),(19,41,31),(20,42,32)], [(1,5,39),(2,40,6),(3,7,37),(4,38,8),(9,36,22),(10,23,33),(11,34,24),(12,21,35),(13,27,45),(14,46,28),(15,25,47),(16,48,26),(17,29,43),(18,44,30),(19,31,41),(20,42,32)], [(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),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,13,29,23),(2,14,30,24),(3,15,31,21),(4,16,32,22),(5,27,43,33),(6,28,44,34),(7,25,41,35),(8,26,42,36),(9,38,48,20),(10,39,45,17),(11,40,46,18),(12,37,47,19)]])
 

60 conjugacy classes

class 1 2A2B2C3A3B3C3D···3H4A···4L6A···6I6J···6X12A···12L
order12223333···34···46···66···612···12
size11112224···49···92···24···418···18

60 irreducible representations

dim111222244444
type+++-++-+
imageC1C2C4S3Dic3D6C4×S3S32S3×Dic3C6.D6C32⋊4D6C33⋊9(C2×C4)
kernelC33⋊6C42C6×C3⋊Dic3C3×C3⋊Dic3C2×C3⋊Dic3C3⋊Dic3C62C3×C6C2×C6C6C6C22C2
# reps13123631236326

Matrix representation of C33⋊6C42 ►in GL6(𝔽13)

100000
010000
001000
000100
00001212
000010
,
12120000
100000
001000
000100
000010
000001
,
100000
010000
0001200
0011200
000010
000001
,
800000
080000
000800
008000
000080
000055
,
1200000
110000
0012000
0001200
000050
000088

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,5,0,0,0,0,0,5],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,8,0,0,0,0,0,8] >;
 

C33⋊6C42 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_6C_4^2
 
% in TeX
 
G:=Group("C3^3:6C4^2");
 
// GroupNames label
 
G:=SmallGroup(432,460);
 
// by ID
 
G=gap.SmallGroup(432,460);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,36,1124,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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