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G = He3.D6order 324 = 22·34

1st non-split extension by He3 of D6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He3.1D6, C9⋊S32S3, (C3×C9)⋊2D6, C32.2S32, He3⋊C2.S3, He3.C3⋊C22, He3.S3⋊C2, He3.C6⋊C2, He3.3S3⋊C2, C3.3(C32⋊D6), SmallGroup(324,40)

Series: Derived Chief Lower central Upper central

C1C32He3.C3 — He3.D6
C1C3C32He3He3.C3He3.C6 — He3.D6
He3.C3 — He3.D6
C1

Generators and relations for He3.D6
 G = < a,b,c,d,e | a3=b3=c3=e2=1, d6=ebe=b-1, ab=ba, cac-1=eae=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=ece=a-1c-1, ede=bd5 >

9C2
27C2
27C2
3C3
9C3
81C22
3S3
9S3
9S3
9C6
9S3
27C6
27S3
27C6
27S3
3C32
3C9
6C9
27D6
27D6
27D6
3C3⋊S3
3D9
3C3⋊S3
3C3×S3
6D9
9D9
9C3×S3
9C3×S3
9C18
9C3×S3
23- 1+2
9S32
9S32
9D18
3C32⋊C6
3C3×D9
3C32⋊C6
3S3×C9
6C9⋊C6
3C32⋊D6
3S3×D9

Character table of He3.D6

 class 12A2B2C3A3B3C6A6B6C9A9B9C9D18A18B18C
 size 192727261818545466636181818
ρ111111111111111111    trivial
ρ21-1-11111-11-11111-1-1-1    linear of order 2
ρ31-11-1111-1-111111-1-1-1    linear of order 2
ρ411-1-11111-1-11111111    linear of order 2
ρ5202022-100-1222-1000    orthogonal lifted from S3
ρ62200222200-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ720-2022-1001222-1000    orthogonal lifted from D6
ρ82-200222-200-1-1-1-1111    orthogonal lifted from D6
ρ9400044-2000-2-2-21000    orthogonal lifted from S32
ρ1060026-300-100000000    orthogonal lifted from C32⋊D6
ρ11600-26-300100000000    orthogonal lifted from C32⋊D6
ρ126-200-3001009594929ζ989492+2ζ9ζ989794+2ζ92098997929594    orthogonal faithful
ρ136200-300-1009594929ζ989492+2ζ9ζ989794+2ζ920ζ989ζ9792ζ9594    orthogonal faithful
ρ146200-300-100ζ989794+2ζ929594929ζ989492+2ζ90ζ9594ζ989ζ9792    orthogonal faithful
ρ156-200-300100ζ989492+2ζ9ζ989794+2ζ929594929097929594989    orthogonal faithful
ρ166-200-300100ζ989794+2ζ929594929ζ989492+2ζ9095949899792    orthogonal faithful
ρ176200-300-100ζ989492+2ζ9ζ989794+2ζ9295949290ζ9792ζ9594ζ989    orthogonal faithful

Permutation representations of He3.D6
On 27 points - transitive group 27T124
Generators in S27
(1 13 10)(2 23 20)(3 15 12)(4 25 22)(5 17 14)(6 27 24)(7 19 16)(8 11 26)(9 21 18)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)
(1 13 22)(2 20 11)(4 25 16)(5 14 23)(7 19 10)(8 26 17)(12 18 24)(15 27 21)
(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 7)(2 6)(3 5)(8 9)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(27,124);

On 27 points - transitive group 27T132
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(11 23 17)(13 25 19)(15 27 21)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)
(1 15 24)(2 19 10)(3 23 14)(4 27 18)(5 13 22)(6 17 26)(7 21 12)(8 25 16)(9 11 20)
(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)(2 3)(5 9)(6 8)(10 17)(11 16)(12 15)(13 14)(18 27)(19 26)(20 25)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(27,132);

Matrix representation of He3.D6 in GL6(𝔽19)

001000
000100
000010
000001
100000
010000
,
1810000
1800000
0018100
0018000
0000181
0000180
,
0000181
0000180
100000
010000
0001800
0011800
,
184416184
15331153
18416181618
153115115
4164161618
3131115
,
1641641816
1313151
164418418
13315315
18164181816
151315151

G:=sub<GL(6,GF(19))| [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],[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],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,18,18,18,18,0,0,0,0,1,0,0,0,0,0],[18,15,18,15,4,3,4,3,4,3,16,1,4,3,16,1,4,3,16,1,18,15,16,1,18,15,16,1,16,1,4,3,18,15,18,15],[16,1,16,1,18,15,4,3,4,3,16,1,16,1,4,3,4,3,4,3,18,15,18,15,18,15,4,3,18,15,16,1,18,15,16,1] >;

He3.D6 in GAP, Magma, Sage, TeX

{\rm He}_3.D_6
% in TeX

G:=Group("He3.D6");
// GroupNames label

G:=SmallGroup(324,40);
// by ID

G=gap.SmallGroup(324,40);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1052,986,579,303,5404,1090,382,3899]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^6=e*b*e=b^-1,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,d*c*d^-1=e*c*e=a^-1*c^-1,e*d*e=b*d^5>;
// generators/relations

Export

Subgroup lattice of He3.D6 in TeX
Character table of He3.D6 in TeX

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