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G = He3.(C3×C6)  order 486 = 2·35

6th non-split extension by He3 of C3×C6 acting via C3×C6/C3=C6

non-abelian, supersoluble, monomial

Aliases: (C3×He3).9C6, He3.6(C3×C6), He3⋊C3⋊3C6, C33.17(C3×S3), He3.2C6⋊1C3, He3⋊C2⋊3C32, He3⋊C32⋊3C2, C32.7(S3×C32), C32.23(C32⋊C6), (C3×3- 1+2)⋊16S3, (C3×C9)⋊7(C3×S3), (C3×He3⋊C2)⋊4C3, C3.21(C3×C32⋊C6), SmallGroup(486,130)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — He3 — He3.(C3×C6)
Chief series
C1 — C3 — C32 — He3 — C3×He3 — He3⋊C32 — He3.(C3×C6)
Lower central
He3 — He3.(C3×C6)
Upper central
C1 — C3 — C32

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

Subgroups: 468 in 82 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×C6, C3×C9, He3, He3, 3- 1+2, C33, C33, S3×C9, He3⋊C2, C2×3- 1+2, S3×C32, He3⋊C3, He3⋊C3, C3×He3, C3×He3, C3×3- 1+2, He3.2C6, S3×3- 1+2, C3×He3⋊C2, He3⋊C32, He3.(C3×C6)
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, He3.(C3×C6)

Permutation representations of He3.(C3×C6)
►On 27 points - transitive group 27T183
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 12)(2 27 10)(3 25 17)(4 23 15)(5 21 13)(6 19 11)(7 26 18)(8 24 16)(9 22 14)

(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 5 8)(3 9 6)(10 21 16 27 13 24)(11 25 14 19 17 22)(12 20)(15 23)(18 26)
 
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), (1,20,12)(2,27,10)(3,25,17)(4,23,15)(5,21,13)(6,19,11)(7,26,18)(8,24,16)(9,22,14), (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), (2,5,8)(3,9,6)(10,21,16,27,13,24)(11,25,14,19,17,22)(12,20)(15,23)(18,26)>;
 
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), (1,20,12)(2,27,10)(3,25,17)(4,23,15)(5,21,13)(6,19,11)(7,26,18)(8,24,16)(9,22,14), (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), (2,5,8)(3,9,6)(10,21,16,27,13,24)(11,25,14,19,17,22)(12,20)(15,23)(18,26) );
 
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)], [(1,20,12),(2,27,10),(3,25,17),(4,23,15),(5,21,13),(6,19,11),(7,26,18),(8,24,16),(9,22,14)], [(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)], [(2,5,8),(3,9,6),(10,21,16,27,13,24),(11,25,14,19,17,22),(12,20),(15,23),(18,26)]])
 
G:=TransitiveGroup(27,183);
 

►On 27 points - transitive group 27T203
Generators in S27
(1 20 14)(2 21 15)(3 22 16)(4 23 17)(5 24 18)(6 25 10)(7 26 11)(8 27 12)(9 19 13)

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

(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 5 8)(3 9 6)(10 19 13 22 16 25)(11 23)(12 27 18 24 15 21)(14 26)(17 20)
 
G:=sub<Sym(27)| (1,20,14)(2,21,15)(3,22,16)(4,23,17)(5,24,18)(6,25,10)(7,26,11)(8,27,12)(9,19,13), (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), (2,15,27)(3,22,10)(5,18,21)(6,25,13)(8,12,24)(9,19,16)(11,17,14)(20,23,26), (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), (2,5,8)(3,9,6)(10,19,13,22,16,25)(11,23)(12,27,18,24,15,21)(14,26)(17,20)>;
 
G:=Group( (1,20,14)(2,21,15)(3,22,16)(4,23,17)(5,24,18)(6,25,10)(7,26,11)(8,27,12)(9,19,13), (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), (2,15,27)(3,22,10)(5,18,21)(6,25,13)(8,12,24)(9,19,16)(11,17,14)(20,23,26), (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), (2,5,8)(3,9,6)(10,19,13,22,16,25)(11,23)(12,27,18,24,15,21)(14,26)(17,20) );
 
G=PermutationGroup([[(1,20,14),(2,21,15),(3,22,16),(4,23,17),(5,24,18),(6,25,10),(7,26,11),(8,27,12),(9,19,13)], [(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)], [(2,15,27),(3,22,10),(5,18,21),(6,25,13),(8,12,24),(9,19,16),(11,17,14),(20,23,26)], [(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)], [(2,5,8),(3,9,6),(10,19,13,22,16,25),(11,23),(12,27,18,24,15,21),(14,26),(17,20)]])
 
G:=TransitiveGroup(27,203);
 

34 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H···3P6A6B6C6D9A···9F18A···18F
order1233333333···366669···918···18
size19113366618···189927279···927···27

34 irreducible representations

dim111111222669
type++++
imageC1C2C3C3C6C6S3C3×S3C3×S3C32⋊C6C3×C32⋊C6He3.(C3×C6)
kernelHe3.(C3×C6)He3⋊C32He3.2C6C3×He3⋊C2He3⋊C3C3×He3C3×3- 1+2C3×C9C33C32C3C1
# reps116262162124

Matrix representation of He3.(C3×C6) ►in GL9(𝔽19)

010000000
001000000
100000000
000010000
000001000
000100000
000000010
000000001
000000100
,
700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
100000000
0110000000
007000000
0000110000
000007000
000100000
000000007
000000100
0000000110
,
000100000
000010000
000001000
000000100
000000010
000000001
700000000
070000000
007000000
,
100000000
007000000
0110000000
0001100000
000001000
000070000
000000700
0000000011
000000010

G:=sub<GL(9,GF(19))| [0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0],[0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0] >;
 

He3.(C3×C6) in GAP, Magma, Sage, TeX 

{\rm He}_3.(C_3\times C_6)
 
% in TeX
 
G:=Group("He3.(C3xC6)");
 
// GroupNames label
 
G:=SmallGroup(486,130);
 
// by ID
 
G=gap.SmallGroup(486,130);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1520,867,873,8104,382]);
 
// 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=a*b^-1,a*d=d*a,e*a*e^-1=a^-1*b,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^-1*b*c,e*c*e^-1=c^-1,e*d*e^-1=b^-1*d>;
 
// generators/relations
 

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