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G = He3.(C3×S3)  order 486 = 2·35

5th non-split extension by He3 of C3×S3 acting via C3×S3/C3=S3

metabelian, supersoluble, monomial

Aliases: C9⋊S3⋊3C32, He3.2S3⋊C3, He3.5(C3×S3), He3⋊C3⋊1C6, (C3×He3).11S3, C33.18(C3×S3), C33.S3⋊3C3, He3⋊C32⋊1C2, C32.9(C32⋊C6), C32.18(S3×C32), (C3×3- 1+2)⋊4C6, (C3×C9)⋊3(C3×C6), C3.9(C3×C32⋊C6), SmallGroup(486,131)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — He3.(C3×S3)
Chief series
C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — He3⋊C32 — He3.(C3×S3)
Lower central
C3×C9 — He3.(C3×S3)
Upper central
C1

Generators and relations for He3.(C3×S3)
 G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=fbf=b-1, ab=ba, cac-1=ab-1, ad=da, ae=ea, faf=a-1, bc=cb, bd=db, be=eb, cd=dc, ece-1=ab-1c, cf=fc, ede-1=b-1d, df=fd, fef=be2 >

Subgroups: 600 in 82 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, C33, C32⋊C6, C9⋊C6, C9⋊S3, S3×C32, C3×C3⋊S3, He3⋊C3, He3⋊C3, C3×He3, C3×He3, C3×3- 1+2, He3.2S3, C3×C32⋊C6, C33.S3, He3⋊C32, He3.(C3×S3)
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, He3.(C3×S3)

Permutation representations of He3.(C3×S3)
►On 27 points - transitive group 27T175
Generators in S27
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 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 26 11)(2 21 12)(3 25 13)(4 20 14)(5 24 15)(6 19 16)(7 23 17)(8 27 18)(9 22 10)

(2 5 8)(3 9 6)(10 16 13)(12 15 18)(19 25 22)(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 9)(3 8)(4 7)(5 6)(10 12)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 27)
 
G:=sub<Sym(27)| (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (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), (1,26,11)(2,21,12)(3,25,13)(4,20,14)(5,24,15)(6,19,16)(7,23,17)(8,27,18)(9,22,10), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(19,25,22)(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), (2,9)(3,8)(4,7)(5,6)(10,12)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,27)>;
 
G:=Group( (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (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), (1,26,11)(2,21,12)(3,25,13)(4,20,14)(5,24,15)(6,19,16)(7,23,17)(8,27,18)(9,22,10), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(19,25,22)(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), (2,9)(3,8)(4,7)(5,6)(10,12)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,27) );
 
G=PermutationGroup([[(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27)], [(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)], [(1,26,11),(2,21,12),(3,25,13),(4,20,14),(5,24,15),(6,19,16),(7,23,17),(8,27,18),(9,22,10)], [(2,5,8),(3,9,6),(10,16,13),(12,15,18),(19,25,22),(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)], [(2,9),(3,8),(4,7),(5,6),(10,12),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,27)]])
 
G:=TransitiveGroup(27,175);
 

►On 27 points - transitive group 27T206
Generators in S27
(1 21 17)(2 22 18)(3 23 10)(4 24 11)(5 25 12)(6 26 13)(7 27 14)(8 19 15)(9 20 16)

(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)

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

(2 5 8)(3 9 6)(10 16 13)(12 15 18)(19 22 25)(20 26 23)

(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 9)(3 8)(4 7)(5 6)(10 19)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)
 
G:=sub<Sym(27)| (1,21,17)(2,22,18)(3,23,10)(4,24,11)(5,25,12)(6,26,13)(7,27,14)(8,19,15)(9,20,16), (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), (2,22,15)(3,10,26)(5,25,18)(6,13,20)(8,19,12)(9,16,23)(11,14,17)(21,27,24), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(19,22,25)(20,26,23), (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), (2,9)(3,8)(4,7)(5,6)(10,19)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)>;
 
G:=Group( (1,21,17)(2,22,18)(3,23,10)(4,24,11)(5,25,12)(6,26,13)(7,27,14)(8,19,15)(9,20,16), (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), (2,22,15)(3,10,26)(5,25,18)(6,13,20)(8,19,12)(9,16,23)(11,14,17)(21,27,24), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(19,22,25)(20,26,23), (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), (2,9)(3,8)(4,7)(5,6)(10,19)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20) );
 
G=PermutationGroup([[(1,21,17),(2,22,18),(3,23,10),(4,24,11),(5,25,12),(6,26,13),(7,27,14),(8,19,15),(9,20,16)], [(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)], [(2,22,15),(3,10,26),(5,25,18),(6,13,20),(8,19,12),(9,16,23),(11,14,17),(21,27,24)], [(2,5,8),(3,9,6),(10,16,13),(12,15,18),(19,22,25),(20,26,23)], [(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)], [(2,9),(3,8),(4,7),(5,6),(10,19),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20)]])
 
G:=TransitiveGroup(27,206);
 

31 conjugacy classes

class 1  2 3A3B3C3D3E3F3G···3L3M···3R6A···6H9A9B9C
order123333333···33···36···6999
size1272336669···918···1827···27181818

31 irreducible representations

dim1111111822266
type+++++
imageC1C2C3C3C6C6He3.(C3×S3)S3C3×S3C3×S3C32⋊C6C3×C32⋊C6
kernelHe3.(C3×S3)He3⋊C32He3.2S3C33.S3He3⋊C3C3×3- 1+2C1C3×He3He3C33C32C3
# reps116262116212

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

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

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

He3.(C3×S3) in GAP, Magma, Sage, TeX 

{\rm He}_3.(C_3\times S_3)
 
% in TeX
 
G:=Group("He3.(C3xS3)");
 
// GroupNames label
 
G:=SmallGroup(486,131);
 
// by ID
 
G=gap.SmallGroup(486,131);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,873,735,453,3244,3250,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=f^2=1,e^3=f*b*f=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a*b^-1*c,c*f=f*c,e*d*e^-1=b^-1*d,d*f=f*d,f*e*f=b*e^2>;
 
// generators/relations
 

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