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## G = He3.D6order 324 = 22·34

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

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

 Derived series C1 — C32 — He3.C3 — He3.D6
 Chief series C1 — C3 — C32 — He3 — He3.C3 — He3.C6 — He3.D6
 Lower central He3.C3 — He3.D6
 Upper central 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
3D9
6D9
9D9
9C18
9S32
9S32
9D18

Character table of He3.D6

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 6C 9A 9B 9C 9D 18A 18B 18C size 1 9 27 27 2 6 18 18 54 54 6 6 6 36 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 0 2 0 2 2 -1 0 0 -1 2 2 2 -1 0 0 0 orthogonal lifted from S3 ρ6 2 2 0 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 0 -2 0 2 2 -1 0 0 1 2 2 2 -1 0 0 0 orthogonal lifted from D6 ρ8 2 -2 0 0 2 2 2 -2 0 0 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ9 4 0 0 0 4 4 -2 0 0 0 -2 -2 -2 1 0 0 0 orthogonal lifted from S32 ρ10 6 0 0 2 6 -3 0 0 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊D6 ρ11 6 0 0 -2 6 -3 0 0 1 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊D6 ρ12 6 -2 0 0 -3 0 0 1 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal faithful ρ13 6 2 0 0 -3 0 0 -1 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal faithful ρ14 6 2 0 0 -3 0 0 -1 0 0 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal faithful ρ15 6 -2 0 0 -3 0 0 1 0 0 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal faithful ρ16 6 -2 0 0 -3 0 0 1 0 0 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal faithful ρ17 6 2 0 0 -3 0 0 -1 0 0 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 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)

 0 0 1 0 0 0 0 0 0 1 0 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
,
 18 1 0 0 0 0 18 0 0 0 0 0 0 0 18 1 0 0 0 0 18 0 0 0 0 0 0 0 18 1 0 0 0 0 18 0
,
 0 0 0 0 18 1 0 0 0 0 18 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 0 0 0 0 1 18 0 0
,
 18 4 4 16 18 4 15 3 3 1 15 3 18 4 16 18 16 18 15 3 1 15 1 15 4 16 4 16 16 18 3 1 3 1 1 15
,
 16 4 16 4 18 16 1 3 1 3 15 1 16 4 4 18 4 18 1 3 3 15 3 15 18 16 4 18 18 16 15 1 3 15 15 1

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

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