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G = C3.He3⋊C6  order 486 = 2·35

The semidirect product of C3.He3 and C6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C3.He3⋊C6, C32.C33⋊C2, C33.18(C3⋊S3), 3- 1+2.S3⋊2C3, (C3×3- 1+2).8S3, 3- 1+2.2(C3×S3), C32.11(He3⋊C2), (C3×C9).8(C3×S3), C32.11(C3×C3⋊S3), C3.12(C3×He3⋊C2), SmallGroup(486,179)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3.He3 — C3.He3⋊C6
Chief series
C1 — C3 — C32 — C3×C9 — C3.He3 — C32.C33 — C3.He3⋊C6
Lower central
C3.He3 — C3.He3⋊C6
Upper central
C1

Generators and relations for C3.He3⋊C6
 G = < a,b,c,d,e | a3=c3=e6=1, b3=eae-1=a-1, d3=a, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=abc-1, ebe-1=b2, dcd-1=ece-1=a-1c, ede-1=a-1d2 >

Subgroups: 416 in 88 conjugacy classes, 19 normal (10 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×D9, C9⋊C6, S3×C32, C3.He3, C3.He3, C3×3- 1+2, C3×3- 1+2, 3- 1+2.S3, C3×C9⋊C6, C32.C33, C3.He3⋊C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, C3×He3⋊C2, C3.He3⋊C6

Permutation representations of C3.He3⋊C6
►On 27 points - transitive group 27T163
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)

(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)

(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)

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

(2 6 8 9 5 3)(4 7)(10 24 13 21 16 27)(11 20)(12 25 18 19 15 22)(14 26)(17 23)
 
G:=sub<Sym(27)| (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (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), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,17,20,7,14,26,4,11,23)(2,15,27,8,12,24,5,18,21)(3,13,25,9,10,22,6,16,19), (2,6,8,9,5,3)(4,7)(10,24,13,21,16,27)(11,20)(12,25,18,19,15,22)(14,26)(17,23)>;
 
G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (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), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,17,20,7,14,26,4,11,23)(2,15,27,8,12,24,5,18,21)(3,13,25,9,10,22,6,16,19), (2,6,8,9,5,3)(4,7)(10,24,13,21,16,27)(11,20)(12,25,18,19,15,22)(14,26)(17,23) );
 
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24)], [(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)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,17,20,7,14,26,4,11,23),(2,15,27,8,12,24,5,18,21),(3,13,25,9,10,22,6,16,19)], [(2,6,8,9,5,3),(4,7),(10,24,13,21,16,27),(11,20),(12,25,18,19,15,22),(14,26),(17,23)]])
 
G:=TransitiveGroup(27,163);
 

31 conjugacy classes

class 1  2 3A3B···3I6A···6H9A···9L
order1233···36···69···9
size12723···327···2718···18

31 irreducible representations

dim1111182223
type++++
imageC1C2C3C6C3.He3⋊C6S3C3×S3C3×S3He3⋊C2
kernelC3.He3⋊C6C32.C333- 1+2.S3C3.He3C1C3×3- 1+2C3×C93- 1+2C32
# reps1122142612

Matrix representation of C3.He3⋊C6 ►in GL18(ℤ)

010000000000000000
-1-10000000000000000
000100000000000000
00-1-100000000000000
000001000000000000
0000-1-1000000000000
000000010000000000
000000-1-10000000000
000000000100000000
00000000-1-100000000
000000000001000000
0000000000-1-1000000
000000000000010000
000000000000-1-10000
000000000000000100
00000000000000-1-100
000000000000000001
0000000000000000-1-1
,
000100000000000000
00-1-100000000000000
000001000000000000
0000-1-1000000000000
100000000000000000
010000000000000000
000000001000000000
000000000100000000
000000000010000000
000000000001000000
000000-1-10000000000
000000100000000000
000000000000001000
000000000000000100
000000000000000010
000000000000000001
000000000000-1-10000
000000000000100000
,
-1-10000000000000000
100000000000000000
00-1-100000000000000
001000000000000000
0000-1-1000000000000
000010000000000000
000000010000000000
000000-1-10000000000
000000000100000000
00000000-1-100000000
000000000001000000
0000000000-1-1000000
000000000000100000
000000000000010000
000000000000001000
000000000000000100
000000000000000010
000000000000000001
,
000000000000100000
000000000000010000
000000000000001000
000000000000000100
000000000000000010
000000000000000001
-1-10000000000000000
100000000000000000
00-1-100000000000000
001000000000000000
0000-1-1000000000000
000010000000000000
000000-1-10000000000
000000100000000000
00000000-1-100000000
000000001000000000
0000000000-1-1000000
000000000010000000
,
100000000000000000
-1-10000000000000000
000010000000000000
0000-1-1000000000000
00-1-100000000000000
000100000000000000
000000000000010000
000000000000100000
000000000000000001
000000000000000010
000000000000001000
00000000000000-1-100
000000010000000000
000000100000000000
000000000001000000
000000000010000000
000000001000000000
00000000-1-100000000

G:=sub<GL(18,Integers())| [0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0] >;
 

C3.He3⋊C6 in GAP, Magma, Sage, TeX 

C_3.{\rm He}_3\rtimes C_6
 
% in TeX
 
G:=Group("C3.He3:C6");
 
// GroupNames label
 
G:=SmallGroup(486,179);
 
// by ID
 
G=gap.SmallGroup(486,179);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,986,4755,2169,303,453,11344,1096,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=c^3=e^6=1,b^3=e*a*e^-1=a^-1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a*b*c^-1,e*b*e^-1=b^2,d*c*d^-1=e*c*e^-1=a^-1*c,e*d*e^-1=a^-1*d^2>;
 
// generators/relations
 

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