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

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

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

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

G:=TransitiveGroup(27,124);

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

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

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

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

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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