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G = He3.2D6  order 324 = 22·34

2nd non-split extension by He3 of D6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He3.2D6, C9⋊S3⋊3S3, (C3×C9)⋊3D6, C32.3S32, He3⋊S3⋊C2, He3⋊C2⋊2S3, He3.2S3⋊C2, He3.2C6⋊C2, He3⋊C3⋊C22, C3.4(C32⋊D6), SmallGroup(324,41)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — He3⋊C3 — He3.2D6
Chief series
C1 — C3 — C32 — He3 — He3⋊C3 — He3.2C6 — He3.2D6
Lower central
He3⋊C3 — He3.2D6
Upper central
C1

Generators and relations for He3.2D6
 G = < a,b,c,d,e | a3=b3=c3=e2=1, d6=b, ab=ba, cac-1=eae=ab-1, dad-1=a-1b, bc=cb, bd=db, ebe=b-1, dcd-1=ece=a-1c-1, ede=b-1d5 >

Subgroups: 596 in 56 conjugacy classes, 12 normal (all characteristic)
C1, C2, C3, C3, C22, S3, C6, C9, C32, C32, D6, D9, C18, C3×S3, C3⋊S3, C3×C9, He3, He3, D18, S32, C3×D9, S3×C9, C32⋊C6, C9⋊S3, He3⋊C2, He3⋊C3, S3×D9, C32⋊D6, He3.2C6, He3.2S3, He3⋊S3, He3.2D6
Quotients: C1, C2, C22, S3, D6, S32, C32⋊D6, He3.2D6

Character table of He3.2D6

 class 12A2B2C3A3B3C3D6A6B6C9A9B9C18A18B18C
 size 192727261836185454666181818
ρ111111111111111111    trivial
ρ21-11-11111-1-11111-1-1-1    linear of order 2
ρ311-1-111111-1-1111111    linear of order 2
ρ41-1-111111-11-1111-1-1-1    linear of order 2
ρ5202022-1-100-1222000    orthogonal lifted from S3
ρ62-200222-1-200-1-1-1111    orthogonal lifted from D6
ρ720-2022-1-1001222000    orthogonal lifted from D6
ρ82200222-1200-1-1-1-1-1-1    orthogonal lifted from S3
ρ9400044-21000-2-2-2000    orthogonal lifted from S32
ρ1060026-3000-10000000    orthogonal lifted from C32⋊D6
ρ11600-26-300010000000    orthogonal lifted from C32⋊D6
ρ126200-3000-100-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9ζ97+ζ92ζ98+ζ9ζ95+ζ94    orthogonal faithful
ρ136-200-3000100ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9-ζ95-ζ94-ζ97-ζ92-ζ98-ζ9    orthogonal faithful
ρ146200-3000-100ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9ζ95+ζ94ζ97+ζ92ζ98+ζ9    orthogonal faithful
ρ156200-3000-1002ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92ζ98+ζ9ζ95+ζ94ζ97+ζ92    orthogonal faithful
ρ166-200-3000100-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9-ζ97-ζ92-ζ98-ζ9-ζ95-ζ94    orthogonal faithful
ρ176-200-30001002ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92-ζ98-ζ9-ζ95-ζ94-ζ97-ζ92    orthogonal faithful

Permutation representations of He3.2D6
►On 27 points - transitive group 27T123
Generators in S27
(1 11 14)(2 21 24)(3 13 16)(4 23 26)(5 15 18)(6 25 10)(7 17 20)(8 27 12)(9 19 22)

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

(1 14 23)(3 13 22)(4 26 17)(6 25 16)(7 20 11)(9 19 10)(12 24 18)(15 21 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 5)(2 4)(6 9)(7 8)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)
 
G:=sub<Sym(27)| (1,11,14)(2,21,24)(3,13,16)(4,23,26)(5,15,18)(6,25,10)(7,17,20)(8,27,12)(9,19,22), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (1,14,23)(3,13,22)(4,26,17)(6,25,16)(7,20,11)(9,19,10)(12,24,18)(15,21,27), (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), (1,5)(2,4)(6,9)(7,8)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)>;
 
G:=Group( (1,11,14)(2,21,24)(3,13,16)(4,23,26)(5,15,18)(6,25,10)(7,17,20)(8,27,12)(9,19,22), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (1,14,23)(3,13,22)(4,26,17)(6,25,16)(7,20,11)(9,19,10)(12,24,18)(15,21,27), (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), (1,5)(2,4)(6,9)(7,8)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20) );
 
G=PermutationGroup([[(1,11,14),(2,21,24),(3,13,16),(4,23,26),(5,15,18),(6,25,10),(7,17,20),(8,27,12),(9,19,22)], [(1,7,4),(2,8,5),(3,9,6),(10,16,22),(11,17,23),(12,18,24),(13,19,25),(14,20,26),(15,21,27)], [(1,14,23),(3,13,22),(4,26,17),(6,25,16),(7,20,11),(9,19,10),(12,24,18),(15,21,27)], [(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)], [(1,5),(2,4),(6,9),(7,8),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20)]])
 
G:=TransitiveGroup(27,123);
 

►On 27 points - transitive group 27T133
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(10 16 22)(12 18 24)(14 20 26)

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

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

(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 3)(4 9)(5 8)(6 7)(10 11)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)
 
G:=sub<Sym(27)| (1,4,7)(2,5,8)(3,6,9)(10,16,22)(12,18,24)(14,20,26), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (1,14,23)(2,12,21)(3,10,19)(4,26,17)(5,24,15)(6,22,13)(7,20,11)(8,18,27)(9,16,25), (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), (1,3)(4,9)(5,8)(6,7)(10,11)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)>;
 
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,22)(12,18,24)(14,20,26), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (1,14,23)(2,12,21)(3,10,19)(4,26,17)(5,24,15)(6,22,13)(7,20,11)(8,18,27)(9,16,25), (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), (1,3)(4,9)(5,8)(6,7)(10,11)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20) );
 
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,16,22),(12,18,24),(14,20,26)], [(1,7,4),(2,8,5),(3,9,6),(10,16,22),(11,17,23),(12,18,24),(13,19,25),(14,20,26),(15,21,27)], [(1,14,23),(2,12,21),(3,10,19),(4,26,17),(5,24,15),(6,22,13),(7,20,11),(8,18,27),(9,16,25)], [(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)], [(1,3),(4,9),(5,8),(6,7),(10,11),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,20)]])
 
G:=TransitiveGroup(27,133);
 

Matrix representation of He3.2D6 ►in GL6(𝔽19)

0180000
1180000
001000
000100
0000181
0000180
,
1810000
1800000
0018100
0018000
0000181
0000180
,
001000
000100
000010
000001
100000
010000
,
7140000
520000
00001417
0000212
00141700
0021200
,
1250000
1770000
000052
0000714
005200
0071400

G:=sub<GL(6,GF(19))| [0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[7,5,0,0,0,0,14,2,0,0,0,0,0,0,0,0,14,2,0,0,0,0,17,12,0,0,14,2,0,0,0,0,17,12,0,0],[12,17,0,0,0,0,5,7,0,0,0,0,0,0,0,0,5,7,0,0,0,0,2,14,0,0,5,7,0,0,0,0,2,14,0,0] >;
 

He3.2D6 in GAP, Magma, Sage, TeX 

{\rm He}_3._2D_6
 
% in TeX
 
G:=Group("He3.2D6");
 
// GroupNames label
 
G:=SmallGroup(324,41);
 
// by ID
 
G=gap.SmallGroup(324,41);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2024,500,579,303,5404,1090,382,3899]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^6=b,a*b=b*a,c*a*c^-1=e*a*e=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=e*c*e=a^-1*c^-1,e*d*e=b^-1*d^5>;
 
// generators/relations
 

Export

Character table of He3.2D6 in TeX

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