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## G = He3.6D6order 324 = 22·34

### The non-split extension by He3 of D6 acting via D6/C3=C22

Aliases: He3.6D6, C9.2S32, C9⋊S34S3, (C3×C9)⋊4D6, C32.4S32, C9○He3⋊C22, He3.4C6⋊C2, He3.4S3⋊C2, He3⋊C2.3S3, C3.3(C324D6), SmallGroup(324,125)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C9○He3 — He3.6D6
 Chief series C1 — C3 — C32 — C3×C9 — C9○He3 — He3.4C6 — He3.6D6
 Lower central C9○He3 — He3.6D6
 Upper central C1

Generators and relations for He3.6D6
G = < a,b,c,d,e | a3=b3=c3=e2=1, d6=ebe=b-1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, bd=db, dcd-1=c-1, ce=ec, ede=bd5 >

Subgroups: 555 in 71 conjugacy classes, 16 normal (8 characteristic)
C1, C2, C3, C3, C22, S3, C6, C9, C9, C32, C32, D6, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, 3- 1+2, D18, S32, C3×D9, S3×C9, C32⋊C6, C9⋊C6, C9⋊S3, He3⋊C2, C9○He3, S3×D9, C32⋊D6, He3.4S3, He3.4C6, He3.6D6
Quotients: C1, C2, C22, S3, D6, S32, C324D6, He3.6D6

Character table of He3.6D6

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 9A 9B 9C 9D 9E 9F 9G 18A 18B 18C size 1 9 27 27 2 6 6 12 18 54 54 2 2 2 12 12 12 12 18 18 18 ρ1 1 1 1 1 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 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 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 1 -1 -1 -1 linear of order 2 ρ5 2 -2 0 0 2 2 2 2 -2 0 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ6 2 0 0 -2 2 -1 2 -1 0 0 1 2 2 2 -1 -1 2 -1 0 0 0 orthogonal lifted from D6 ρ7 2 0 -2 0 2 2 -1 -1 0 1 0 2 2 2 -1 2 -1 -1 0 0 0 orthogonal lifted from D6 ρ8 2 0 0 2 2 -1 2 -1 0 0 -1 2 2 2 -1 -1 2 -1 0 0 0 orthogonal lifted from S3 ρ9 2 2 0 0 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 0 2 0 2 2 -1 -1 0 -1 0 2 2 2 -1 2 -1 -1 0 0 0 orthogonal lifted from S3 ρ11 4 0 0 0 4 4 -2 -2 0 0 0 -2 -2 -2 1 -2 1 1 0 0 0 orthogonal lifted from S32 ρ12 4 0 0 0 4 -2 4 -2 0 0 0 -2 -2 -2 1 1 -2 1 0 0 0 orthogonal lifted from S32 ρ13 4 0 0 0 4 -2 -2 1 0 0 0 4 4 4 1 -2 -2 1 0 0 0 orthogonal lifted from S32 ρ14 4 0 0 0 4 -2 -2 1 0 0 0 -2 -2 -2 -1+3√-3/2 1 1 -1-3√-3/2 0 0 0 complex lifted from C32⋊4D6 ρ15 4 0 0 0 4 -2 -2 1 0 0 0 -2 -2 -2 -1-3√-3/2 1 1 -1+3√-3/2 0 0 0 complex lifted from C32⋊4D6 ρ16 6 2 0 0 -3 0 0 0 -1 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal faithful ρ17 6 2 0 0 -3 0 0 0 -1 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal faithful ρ18 6 -2 0 0 -3 0 0 0 1 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal faithful ρ19 6 -2 0 0 -3 0 0 0 1 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal faithful ρ20 6 2 0 0 -3 0 0 0 -1 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal faithful ρ21 6 -2 0 0 -3 0 0 0 1 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal faithful

Permutation representations of He3.6D6
On 27 points - transitive group 27T122
Generators in S27
(1 23 14)(2 15 24)(3 25 16)(4 17 26)(5 27 18)(6 19 10)(7 11 20)(8 21 12)(9 13 22)
(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)
(10 16 22)(11 23 17)(12 18 24)(13 25 19)(14 20 26)(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 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)| (1,23,14)(2,15,24)(3,25,16)(4,17,26)(5,27,18)(6,19,10)(7,11,20)(8,21,12)(9,13,22), (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), (10,16,22)(11,23,17)(12,18,24)(13,25,19)(14,20,26)(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), (1,3)(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( (1,23,14)(2,15,24)(3,25,16)(4,17,26)(5,27,18)(6,19,10)(7,11,20)(8,21,12)(9,13,22), (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), (10,16,22)(11,23,17)(12,18,24)(13,25,19)(14,20,26)(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), (1,3)(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([[(1,23,14),(2,15,24),(3,25,16),(4,17,26),(5,27,18),(6,19,10),(7,11,20),(8,21,12),(9,13,22)], [(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)], [(10,16,22),(11,23,17),(12,18,24),(13,25,19),(14,20,26),(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)], [(1,3),(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(27,122);

Matrix representation of He3.6D6 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 1 0 0 0 0 18 0 0 0 18 18 0 18 18 18 0 0 1 0 1 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 1 0 1 0 0 1 0 18 0 18 18 18
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 1 18 18 18 18 17 18 0 0 0 0 1 0 1 0 0 0 1 0
,
 12 2 0 0 0 0 17 14 0 0 0 0 17 17 17 17 3 10 5 5 5 5 12 3 2 0 2 0 14 2 2 0 14 2 14 2
,
 18 1 0 0 0 0 0 1 0 0 0 0 0 0 18 1 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 18 0 18 0 18

G:=sub<GL(6,GF(19))| [1,0,0,0,18,0,0,1,0,0,18,0,0,0,18,18,0,1,0,0,1,0,18,0,0,0,0,0,18,1,0,0,0,0,18,0],[18,18,0,0,1,0,1,0,0,0,0,18,0,0,18,18,1,0,0,0,1,0,0,18,0,0,0,0,0,18,0,0,0,0,1,18],[0,0,0,18,0,1,0,0,0,18,0,0,1,0,0,18,0,0,0,1,0,18,0,0,0,0,18,17,1,1,0,0,1,18,0,0],[12,17,17,5,2,2,2,14,17,5,0,0,0,0,17,5,2,14,0,0,17,5,0,2,0,0,3,12,14,14,0,0,10,3,2,2],[18,0,0,0,1,0,1,1,0,0,0,18,0,0,18,0,1,0,0,0,1,1,0,18,0,0,0,0,1,0,0,0,0,0,1,18] >;

He3.6D6 in GAP, Magma, Sage, TeX

{\rm He}_3._6D_6
% in TeX

G:=Group("He3.6D6");
// GroupNames label

G:=SmallGroup(324,125);
// by ID

G=gap.SmallGroup(324,125);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,80,1593,453,2164,1096,3899]);
// 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=a*b^-1,d*a*d^-1=e*a*e=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=b*d^5>;
// generators/relations

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