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G = C34:6S3  order 486 = 2·35

6th semidirect product of C34 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C34:6S3, C32:He3:5C2, C33:2(C3:S3), (C3xHe3):14S3, C3.8(He3:5S3), C32:2(He3:C2), C32.15(C33:C2), SmallGroup(486,183)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C32:He3 — C34:6S3
Chief series
C1 — C3 — C32 — C33 — C3xHe3 — C32:He3 — C34:6S3
Lower central
C32:He3 — C34:6S3
Upper central
C1 — C32

Generators and relations for C34:6S3
 G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, eae-1=ad-1, faf=a-1, bc=cb, bd=db, ebe-1=bc-1, fbf=b-1, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1486 in 267 conjugacy classes, 38 normal (5 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C32, C3xS3, C3:S3, C3xC6, He3, C33, C33, He3:C2, S3xC32, C3xC3:S3, C3xHe3, C34, C3xHe3:C2, C32xC3:S3, C32:He3, C34:6S3
Quotients: C1, C2, S3, C3:S3, He3:C2, C33:C2, He3:5S3, C34:6S3

Permutation representations of C34:6S3
►On 27 points - transitive group 27T154
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

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

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

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

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

(10 20)(11 19)(12 21)(13 24)(14 23)(15 22)(16 26)(17 25)(18 27)
 
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (10,13,16)(11,14,17)(12,15,18)(19,25,23)(20,26,24)(21,27,22), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,23)(20,26,24)(21,27,22), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27), (1,10,20)(2,11,19)(3,12,21)(4,13,24)(5,14,23)(6,15,22)(7,16,26)(8,17,25)(9,18,27), (10,20)(11,19)(12,21)(13,24)(14,23)(15,22)(16,26)(17,25)(18,27)>;
 
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (10,13,16)(11,14,17)(12,15,18)(19,25,23)(20,26,24)(21,27,22), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,23)(20,26,24)(21,27,22), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27), (1,10,20)(2,11,19)(3,12,21)(4,13,24)(5,14,23)(6,15,22)(7,16,26)(8,17,25)(9,18,27), (10,20)(11,19)(12,21)(13,24)(14,23)(15,22)(16,26)(17,25)(18,27) );
 
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(10,13,16),(11,14,17),(12,15,18),(19,25,23),(20,26,24),(21,27,22)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,23),(20,26,24),(21,27,22)], [(1,3,2),(4,6,5),(7,9,8),(10,12,11),(13,15,14),(16,18,17),(19,20,21),(22,23,24),(25,26,27)], [(1,10,20),(2,11,19),(3,12,21),(4,13,24),(5,14,23),(6,15,22),(7,16,26),(8,17,25),(9,18,27)], [(10,20),(11,19),(12,21),(13,24),(14,23),(15,22),(16,26),(17,25),(18,27)]])
 
G:=TransitiveGroup(27,154);
 

39 conjugacy classes

class 1  2 3A···3H3I···3T3U···3AC6A···6H
order123···33···33···36···6
size1271···16···618···1827···27

39 irreducible representations

dim112236
type++++
imageC1C2S3S3He3:C2He3:5S3
kernelC34:6S3C32:He3C3xHe3C34C32C3
# reps11121168

Matrix representation of C34:6S3 ►in GL6(F7)

100000
020000
004000
000100
000040
000002
,
100000
040000
002000
000100
000040
000002
,
200000
020000
002000
000200
000020
000002
,
400000
040000
004000
000200
000020
000002
,
001000
100000
010000
000001
000100
000010
,
100000
001000
010000
000100
000001
000010

G:=sub<GL(6,GF(7))| [1,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C34:6S3 in GAP, Magma, Sage, TeX 

C_3^4\rtimes_6S_3
 
% in TeX
 
G:=Group("C3^4:6S3");
 
// GroupNames label
 
G:=SmallGroup(486,183);
 
// by ID
 
G=gap.SmallGroup(486,183);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,867,2169,303]);
 
// 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*d^-1,f*a*f=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^-1,f*b*f=b^-1,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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