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G = He3.C3⋊2C6  order 486 = 2·35

2nd semidirect product of He3.C3 and C6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He3.C3⋊2C6, He3.3S3⋊C3, He3.8(C3×S3), (C3×He3).18S3, C33.16(C3⋊S3), He3.C32⋊2C2, 3- 1+2⋊4(C3×S3), (C3×3- 1+2)⋊8S3, C32.9(He3⋊C2), (C3×C9)⋊2(C3×S3), C32.9(C3×C3⋊S3), C3.10(C3×He3⋊C2), SmallGroup(486,177)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — He3.C3 — He3.C3⋊2C6
Chief series
C1 — C3 — C32 — C3×C9 — He3.C3 — He3.C32 — He3.C3⋊2C6
Lower central
He3.C3 — He3.C3⋊2C6
Upper central
C1

Generators and relations for He3.C3⋊2C6
 G = < a,b,c,d,e | a3=b3=c3=e6=1, d3=b, ab=ba, cac-1=eae-1=ab-1, ad=da, bc=cb, bd=db, ebe-1=b-1, dcd-1=ab-1c, ece-1=c-1, ede-1=bd2 >

Subgroups: 524 in 94 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C3×D9, C32⋊C6, C9⋊C6, S3×C32, C3×C3⋊S3, He3.C3, He3.C3, C3×He3, C3×3- 1+2, C3×3- 1+2, He3.3S3, C3×C32⋊C6, C3×C9⋊C6, He3.C32, He3.C3⋊2C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, C3×He3⋊C2, He3.C3⋊2C6

Permutation representations of He3.C3⋊2C6
►On 27 points - transitive group 27T153
Generators in S27
(1 24 16)(2 25 17)(3 26 18)(4 27 10)(5 19 11)(6 20 12)(7 21 13)(8 22 14)(9 23 15)

(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)

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

(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)

(2 3 5 9 8 6)(4 7)(10 16)(11 18 14 15 17 12)(19 20 22 26 25 23)(21 24)
 
G:=sub<Sym(27)| (1,24,16)(2,25,17)(3,26,18)(4,27,10)(5,19,11)(6,20,12)(7,21,13)(8,22,14)(9,23,15), (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), (2,25,11)(3,18,23)(5,19,14)(6,12,26)(8,22,17)(9,15,20)(10,16,13)(21,24,27), (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), (2,3,5,9,8,6)(4,7)(10,16)(11,18,14,15,17,12)(19,20,22,26,25,23)(21,24)>;
 
G:=Group( (1,24,16)(2,25,17)(3,26,18)(4,27,10)(5,19,11)(6,20,12)(7,21,13)(8,22,14)(9,23,15), (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), (2,25,11)(3,18,23)(5,19,14)(6,12,26)(8,22,17)(9,15,20)(10,16,13)(21,24,27), (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), (2,3,5,9,8,6)(4,7)(10,16)(11,18,14,15,17,12)(19,20,22,26,25,23)(21,24) );
 
G=PermutationGroup([[(1,24,16),(2,25,17),(3,26,18),(4,27,10),(5,19,11),(6,20,12),(7,21,13),(8,22,14),(9,23,15)], [(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)], [(2,25,11),(3,18,23),(5,19,14),(6,12,26),(8,22,17),(9,15,20),(10,16,13),(21,24,27)], [(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)], [(2,3,5,9,8,6),(4,7),(10,16),(11,18,14,15,17,12),(19,20,22,26,25,23),(21,24)]])
 
G:=TransitiveGroup(27,153);
 

►On 27 points - transitive group 27T173
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(19 22 25)(20 23 26)(21 24 27)

(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)

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

(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)

(2 3 5 9 8 6)(4 7)(10 27 16 21 13 24)(11 20)(12 22 15 19 18 25)(14 26)(17 23)
 
G:=sub<Sym(27)| (1,7,4)(2,8,5)(3,9,6)(19,22,25)(20,23,26)(21,24,27), (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), (1,20,11)(2,24,15)(3,19,10)(4,23,14)(5,27,18)(6,22,13)(7,26,17)(8,21,12)(9,25,16), (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), (2,3,5,9,8,6)(4,7)(10,27,16,21,13,24)(11,20)(12,22,15,19,18,25)(14,26)(17,23)>;
 
G:=Group( (1,7,4)(2,8,5)(3,9,6)(19,22,25)(20,23,26)(21,24,27), (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), (1,20,11)(2,24,15)(3,19,10)(4,23,14)(5,27,18)(6,22,13)(7,26,17)(8,21,12)(9,25,16), (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), (2,3,5,9,8,6)(4,7)(10,27,16,21,13,24)(11,20)(12,22,15,19,18,25)(14,26)(17,23) );
 
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(19,22,25),(20,23,26),(21,24,27)], [(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)], [(1,20,11),(2,24,15),(3,19,10),(4,23,14),(5,27,18),(6,22,13),(7,26,17),(8,21,12),(9,25,16)], [(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)], [(2,3,5,9,8,6),(4,7),(10,27,16,21,13,24),(11,20),(12,22,15,19,18,25),(14,26),(17,23)]])
 
G:=TransitiveGroup(27,173);
 

31 conjugacy classes

class 1  2 3A3B···3I3J3K3L6A···6H9A···9I
order1233···33336···69···9
size12723···318181827···2718···18

31 irreducible representations

dim111118222223
type+++++
imageC1C2C3C6He3.C3⋊2C6S3S3C3×S3C3×S3C3×S3He3⋊C2
kernelHe3.C3⋊2C6He3.C32He3.3S3He3.C3C1C3×He3C3×3- 1+2C3×C9He33- 1+2C32
# reps112211322412

Matrix representation of He3.C3⋊2C6 ►in GL18(ℤ)

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

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

He3.C3⋊2C6 in GAP, Magma, Sage, TeX 

{\rm He}_3.C_3\rtimes_2C_6
 
% in TeX
 
G:=Group("He3.C3:2C6");
 
// GroupNames label
 
G:=SmallGroup(486,177);
 
// by ID
 
G=gap.SmallGroup(486,177);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,8643,2169,303,237,11344,1096,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^6=1,d^3=b,a*b=b*a,c*a*c^-1=e*a*e^-1=a*b^-1,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,d*c*d^-1=a*b^-1*c,e*c*e^-1=c^-1,e*d*e^-1=b*d^2>;
 
// generators/relations
 

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