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G = C34⋊S3  order 486 = 2·35

1st semidirect product of C34 and S3 acting faithfully

metabelian, supersoluble, monomial

Aliases: C34⋊1S3, (C3×He3)⋊3C6, C33⋊1(C3×C6), C32⋊He3⋊2C2, He3⋊4S3⋊1C3, C33.33(C3×S3), C32⋊1(C32⋊C6), C33⋊C2⋊1C32, C32.36(S3×C32), C3.2(C3×C32⋊C6), SmallGroup(486,103)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C34⋊S3
Chief series
C1 — C3 — C32 — C33 — C3×He3 — C32⋊He3 — C34⋊S3
Lower central
C33 — C34⋊S3
Upper central
C1

Generators and relations for C34⋊S3
 G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, eae-1=ac-1, af=fa, bc=cb, bd=db, ebe-1=bd-1, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1086 in 171 conjugacy classes, 27 normal (7 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, C3×C6, He3, C33, C33, C33, C32⋊C6, S3×C32, C3×C3⋊S3, C33⋊C2, C3×He3, C3×He3, C34, He3⋊4S3, C32×C3⋊S3, C32⋊He3, C34⋊S3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, C34⋊S3

Permutation representations of C34⋊S3
►On 27 points - transitive group 27T151
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

(10 27 21)(11 25 19)(12 26 20)(13 23 16)(14 24 17)(15 22 18)

(1 7 4)(2 8 5)(3 9 6)(10 11 12)(13 15 14)(16 18 17)(19 20 21)(22 24 23)(25 26 27)

(1 3 2)(4 6 5)(7 9 8)(10 27 21)(11 25 19)(12 26 20)(13 16 23)(14 17 24)(15 18 22)

(1 24 12)(2 17 20)(3 14 26)(4 22 11)(5 18 19)(6 15 25)(7 23 10)(8 16 21)(9 13 27)

(2 3)(4 7)(5 9)(6 8)(10 22)(11 23)(12 24)(13 19)(14 20)(15 21)(16 25)(17 26)(18 27)
 
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (10,27,21)(11,25,19)(12,26,20)(13,23,16)(14,24,17)(15,22,18), (1,7,4)(2,8,5)(3,9,6)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,27,21)(11,25,19)(12,26,20)(13,16,23)(14,17,24)(15,18,22), (1,24,12)(2,17,20)(3,14,26)(4,22,11)(5,18,19)(6,15,25)(7,23,10)(8,16,21)(9,13,27), (2,3)(4,7)(5,9)(6,8)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)(16,25)(17,26)(18,27)>;
 
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (10,27,21)(11,25,19)(12,26,20)(13,23,16)(14,24,17)(15,22,18), (1,7,4)(2,8,5)(3,9,6)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,27,21)(11,25,19)(12,26,20)(13,16,23)(14,17,24)(15,18,22), (1,24,12)(2,17,20)(3,14,26)(4,22,11)(5,18,19)(6,15,25)(7,23,10)(8,16,21)(9,13,27), (2,3)(4,7)(5,9)(6,8)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)(16,25)(17,26)(18,27) );
 
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(10,27,21),(11,25,19),(12,26,20),(13,23,16),(14,24,17),(15,22,18)], [(1,7,4),(2,8,5),(3,9,6),(10,11,12),(13,15,14),(16,18,17),(19,20,21),(22,24,23),(25,26,27)], [(1,3,2),(4,6,5),(7,9,8),(10,27,21),(11,25,19),(12,26,20),(13,16,23),(14,17,24),(15,18,22)], [(1,24,12),(2,17,20),(3,14,26),(4,22,11),(5,18,19),(6,15,25),(7,23,10),(8,16,21),(9,13,27)], [(2,3),(4,7),(5,9),(6,8),(10,22),(11,23),(12,24),(13,19),(14,20),(15,21),(16,25),(17,26),(18,27)]])
 
G:=TransitiveGroup(27,151);
 

39 conjugacy classes

class 1  2 3A3B3C3D3E···3L3M···3T3U···3AC6A···6H
order1233333···33···33···36···6
size12722223···36···618···1827···27

39 irreducible representations

dim11112266
type++++
imageC1C2C3C6S3C3×S3C32⋊C6C3×C32⋊C6
kernelC34⋊S3C32⋊He3He3⋊4S3C3×He3C34C33C32C3
# reps11881848

Matrix representation of C34⋊S3 ►in GL12(ℤ)

100000000000
010000000000
000100000000
00-1-100000000
0000-1-1000000
000010000000
000000100000
000000010000
00000000-1-100
000000001000
000000000001
0000000000-1-1
,
100000000000
010000000000
00-1-100000000
001000000000
000001000000
0000-1-1000000
000000100000
000000010000
00000000-1-100
000000001000
000000000001
0000000000-1-1
,
-1-10000000000
100000000000
00-1-100000000
001000000000
0000-1-1000000
000010000000
000000010000
000000-1-10000
000000000100
00000000-1-100
000000000001
0000000000-1-1
,
010000000000
-1-10000000000
000100000000
00-1-100000000
000001000000
0000-1-1000000
000000010000
000000-1-10000
000000000100
00000000-1-100
000000000001
0000000000-1-1
,
000010000000
000001000000
100000000000
010000000000
001000000000
000100000000
000000000010
000000000001
000000100000
000000010000
000000001000
000000000100
,
0-10000000000
-100000000000
00000-1000000
0000-10000000
000-100000000
00-1000000000
000000-1-10000
000000010000
0000000000-1-1
000000000001
00000000-1-100
000000000100

G:=sub<GL(12,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0] >;
 

C34⋊S3 in GAP, Magma, Sage, TeX 

C_3^4\rtimes S_3
 
% in TeX
 
G:=Group("C3^4:S3");
 
// GroupNames label
 
G:=SmallGroup(486,103);
 
// by ID
 
G=gap.SmallGroup(486,103);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,873,735,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*c^-1,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*d^-1,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f=c^-1,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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