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G = He3.C3⋊C6  order 486 = 2·35

3rd semidirect product of He3.C3 and C6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He3.C3⋊3C6, He3.5(C3×C6), (C3×He3).8C6, C33.15(C3×S3), He3.C6⋊1C3, C32.6(S3×C32), He3⋊C2.4C32, He3.C32⋊3C2, C32.22(C32⋊C6), (C3×3- 1+2)⋊15S3, (C3×C9)⋊6(C3×S3), C3.20(C3×C32⋊C6), (C3×He3⋊C2).3C3, SmallGroup(486,128)

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=ab-1c, ece-1=bc-1, ede-1=b-1d >

Subgroups: 360 in 76 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×C6, C3×C9, 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, C3×3- 1+2, C3×3- 1+2, He3.C6, 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 27T190
Generators in S27
(1 23 13)(2 24 14)(3 25 15)(4 26 16)(5 27 17)(6 19 18)(7 20 10)(8 21 11)(9 22 12)

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

(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 8 5)(10 20 16 26 13 23)(11 24 14 27 17 21)(12 19)(15 22)(18 25)
 
G:=sub<Sym(27)| (1,23,13)(2,24,14)(3,25,15)(4,26,16)(5,27,17)(6,19,18)(7,20,10)(8,21,11)(9,22,12), (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), (1,10,23)(2,8,5)(3,22,12)(4,13,26)(6,25,15)(7,16,20)(9,19,18)(11,14,17), (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)(2,8,5)(10,20,16,26,13,23)(11,24,14,27,17,21)(12,19)(15,22)(18,25)>;
 
G:=Group( (1,23,13)(2,24,14)(3,25,15)(4,26,16)(5,27,17)(6,19,18)(7,20,10)(8,21,11)(9,22,12), (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), (1,10,23)(2,8,5)(3,22,12)(4,13,26)(6,25,15)(7,16,20)(9,19,18)(11,14,17), (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)(2,8,5)(10,20,16,26,13,23)(11,24,14,27,17,21)(12,19)(15,22)(18,25) );
 
G=PermutationGroup([[(1,23,13),(2,24,14),(3,25,15),(4,26,16),(5,27,17),(6,19,18),(7,20,10),(8,21,11),(9,22,12)], [(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)], [(1,10,23),(2,8,5),(3,22,12),(4,13,26),(6,25,15),(7,16,20),(9,19,18),(11,14,17)], [(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),(2,8,5),(10,20,16,26,13,23),(11,24,14,27,17,21),(12,19),(15,22),(18,25)]])
 
G:=TransitiveGroup(27,190);
 

►On 27 points - transitive group 27T204
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)

(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 16 22)(2 11 26)(3 15 21)(4 10 25)(5 14 20)(6 18 24)(7 13 19)(8 17 23)(9 12 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 5 8)(3 9 6)(10 19)(11 23 17 20 14 26)(12 27 15 21 18 24)(13 22)(16 25)
 
G:=sub<Sym(27)| (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (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), (1,16,22)(2,11,26)(3,15,21)(4,10,25)(5,14,20)(6,18,24)(7,13,19)(8,17,23)(9,12,27), (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), (2,5,8)(3,9,6)(10,19)(11,23,17,20,14,26)(12,27,15,21,18,24)(13,22)(16,25)>;
 
G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (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), (1,16,22)(2,11,26)(3,15,21)(4,10,25)(5,14,20)(6,18,24)(7,13,19)(8,17,23)(9,12,27), (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), (2,5,8)(3,9,6)(10,19)(11,23,17,20,14,26)(12,27,15,21,18,24)(13,22)(16,25) );
 
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18)], [(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)], [(1,16,22),(2,11,26),(3,15,21),(4,10,25),(5,14,20),(6,18,24),(7,13,19),(8,17,23),(9,12,27)], [(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)], [(2,5,8),(3,9,6),(10,19),(11,23,17,20,14,26),(12,27,15,21,18,24),(13,22),(16,25)]])
 
G:=TransitiveGroup(27,204);
 

34 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H3I3J6A6B6C6D9A···9F9G···9L18A···18F
order12333333333366669···99···918···18
size1911336661818189927279···918···1827···27

34 irreducible representations

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

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

0110000000
0011000000
1100000000
0000110000
0000011000
0001100000
0000000110
0000000011
0000001100
,
1100000000
0110000000
0011000000
0001100000
0000110000
0000011000
0000001100
0000000110
0000000011
,
007000000
1100000000
010000000
000070000
0000011000
000100000
000000700
0000000110
000000001
,
0001100000
0000110000
0000011000
000000700
000000070
000000007
1100000000
0110000000
0011000000
,
1100000000
001000000
070000000
000100000
000007000
0000110000
000000700
0000000011
000000010

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

He3.C3⋊C6 in GAP, Magma, Sage, TeX 

{\rm He}_3.C_3\rtimes C_6
 
% in TeX
 
G:=Group("He3.C3:C6");
 
// GroupNames label
 
G:=SmallGroup(486,128);
 
// by ID
 
G=gap.SmallGroup(486,128);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1520,867,873,12964,652]);
 
// 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*b^-1*c,e*c*e^-1=b*c^-1,e*d*e^-1=b^-1*d>;
 
// generators/relations
 

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