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G = He3⋊(C3×S3)  order 486 = 2·35

4th semidirect product of He3 and C3×S3 acting via C3×S3/C3=S3

non-abelian, supersoluble, monomial

Aliases: He3⋊4(C3×S3), (C3×He3)⋊13S3, He3⋊C3⋊2C6, He3⋊S3⋊2C3, C33.17(C3⋊S3), He3⋊C32⋊2C2, (C3×3- 1+2)⋊9S3, C32.10(He3⋊C2), (C3×C9)⋊3(C3×S3), C32.10(C3×C3⋊S3), C3.11(C3×He3⋊C2), SmallGroup(486,178)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — He3⋊C3 — He3⋊(C3×S3)
Chief series
C1 — C3 — C32 — He3 — He3⋊C3 — He3⋊C32 — He3⋊(C3×S3)
Lower central
He3⋊C3 — He3⋊(C3×S3)
Upper central
C1

Generators and relations for He3⋊(C3×S3)
 G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, eae-1=ab=ba, cac-1=ab-1, ad=da, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=b-1c, ece-1=a-1c, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 740 in 106 conjugacy classes, 19 normal (10 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, C33, C33, C3×D9, C32⋊C6, C9⋊C6, S3×C32, C3×C3⋊S3, He3⋊C3, He3⋊C3, C3×He3, C3×3- 1+2, He3⋊S3, C3×C32⋊C6, C3×C9⋊C6, He3⋊C32, He3⋊(C3×S3)
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, C3×He3⋊C2, He3⋊(C3×S3)

Permutation representations of He3⋊(C3×S3)
►On 27 points - transitive group 27T147
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

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

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

(2 4 5)(6 7 8)(13 15 14)(19 20 21)(22 24 23)(25 26 27)

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

(2 6)(3 9)(4 7)(5 8)(10 16)(11 17)(12 18)(13 23)(14 24)(15 22)(19 27)(20 25)(21 26)
 
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,9,3)(2,5,4)(6,7,8)(10,12,11)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,26,27), (1,23,13)(2,10,27)(3,24,15)(4,11,26)(5,12,25)(6,19,16)(7,21,17)(8,20,18)(9,22,14), (2,4,5)(6,7,8)(13,15,14)(19,20,21)(22,24,23)(25,26,27), (1,16,10)(2,13,20)(3,18,11)(4,15,21)(5,14,19)(6,25,23)(7,26,22)(8,27,24)(9,17,12), (2,6)(3,9)(4,7)(5,8)(10,16)(11,17)(12,18)(13,23)(14,24)(15,22)(19,27)(20,25)(21,26)>;
 
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,9,3)(2,5,4)(6,7,8)(10,12,11)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,26,27), (1,23,13)(2,10,27)(3,24,15)(4,11,26)(5,12,25)(6,19,16)(7,21,17)(8,20,18)(9,22,14), (2,4,5)(6,7,8)(13,15,14)(19,20,21)(22,24,23)(25,26,27), (1,16,10)(2,13,20)(3,18,11)(4,15,21)(5,14,19)(6,25,23)(7,26,22)(8,27,24)(9,17,12), (2,6)(3,9)(4,7)(5,8)(10,16)(11,17)(12,18)(13,23)(14,24)(15,22)(19,27)(20,25)(21,26) );
 
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,9,3),(2,5,4),(6,7,8),(10,12,11),(13,14,15),(16,17,18),(19,21,20),(22,24,23),(25,26,27)], [(1,23,13),(2,10,27),(3,24,15),(4,11,26),(5,12,25),(6,19,16),(7,21,17),(8,20,18),(9,22,14)], [(2,4,5),(6,7,8),(13,15,14),(19,20,21),(22,24,23),(25,26,27)], [(1,16,10),(2,13,20),(3,18,11),(4,15,21),(5,14,19),(6,25,23),(7,26,22),(8,27,24),(9,17,12)], [(2,6),(3,9),(4,7),(5,8),(10,16),(11,17),(12,18),(13,23),(14,24),(15,22),(19,27),(20,25),(21,26)]])
 
G:=TransitiveGroup(27,147);
 

►On 27 points - transitive group 27T161
Generators in S27
(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 11 13)(2 12 14)(3 10 15)(4 8 26)(5 9 27)(6 7 25)(16 22 19)(17 23 20)(18 24 21)

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

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

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

(4 23)(5 24)(6 22)(7 16)(8 17)(9 18)(10 15)(11 13)(12 14)(19 25)(20 26)(21 27)
 
G:=sub<Sym(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), (1,11,13)(2,12,14)(3,10,15)(4,8,26)(5,9,27)(6,7,25)(16,22,19)(17,23,20)(18,24,21), (2,12,14)(3,15,10)(4,25,5)(6,9,8)(7,27,26)(16,20,21)(17,18,22)(19,23,24), (1,3,2)(4,7,27)(5,8,25)(6,9,26)(10,12,11)(13,15,14)(16,21,23)(17,19,24)(18,20,22), (1,5,24)(2,25,19)(3,8,17)(4,20,15)(6,16,12)(7,22,14)(9,21,11)(10,26,23)(13,27,18), (4,23)(5,24)(6,22)(7,16)(8,17)(9,18)(10,15)(11,13)(12,14)(19,25)(20,26)(21,27)>;
 
G:=Group( (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), (1,11,13)(2,12,14)(3,10,15)(4,8,26)(5,9,27)(6,7,25)(16,22,19)(17,23,20)(18,24,21), (2,12,14)(3,15,10)(4,25,5)(6,9,8)(7,27,26)(16,20,21)(17,18,22)(19,23,24), (1,3,2)(4,7,27)(5,8,25)(6,9,26)(10,12,11)(13,15,14)(16,21,23)(17,19,24)(18,20,22), (1,5,24)(2,25,19)(3,8,17)(4,20,15)(6,16,12)(7,22,14)(9,21,11)(10,26,23)(13,27,18), (4,23)(5,24)(6,22)(7,16)(8,17)(9,18)(10,15)(11,13)(12,14)(19,25)(20,26)(21,27) );
 
G=PermutationGroup([[(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)], [(1,11,13),(2,12,14),(3,10,15),(4,8,26),(5,9,27),(6,7,25),(16,22,19),(17,23,20),(18,24,21)], [(2,12,14),(3,15,10),(4,25,5),(6,9,8),(7,27,26),(16,20,21),(17,18,22),(19,23,24)], [(1,3,2),(4,7,27),(5,8,25),(6,9,26),(10,12,11),(13,15,14),(16,21,23),(17,19,24),(18,20,22)], [(1,5,24),(2,25,19),(3,8,17),(4,20,15),(6,16,12),(7,22,14),(9,21,11),(10,26,23),(13,27,18)], [(4,23),(5,24),(6,22),(7,16),(8,17),(9,18),(10,15),(11,13),(12,14),(19,25),(20,26),(21,27)]])
 
G:=TransitiveGroup(27,161);
 

31 conjugacy classes

class 1  2 3A3B···3I3J···3R6A···6H9A9B9C
order1233···33···36···6999
size12723···318···1827···27181818

31 irreducible representations

dim11111822223
type+++++
imageC1C2C3C6He3⋊(C3×S3)S3S3C3×S3C3×S3He3⋊C2
kernelHe3⋊(C3×S3)He3⋊C32He3⋊S3He3⋊C3C1C3×He3C3×3- 1+2C3×C9He3C32
# reps11221312612

Matrix representation of He3⋊(C3×S3) ►in GL18(ℤ)

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

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

He3⋊(C3×S3) in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes (C_3\times S_3)
 
% in TeX
 
G:=Group("He3:(C3xS3)");
 
// GroupNames label
 
G:=SmallGroup(486,178);
 
// by ID
 
G=gap.SmallGroup(486,178);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,1520,867,303,1096,652,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,e*a*e^-1=a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,d*c*d^-1=b^-1*c,e*c*e^-1=a^-1*c,f*c*f=c^-1,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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