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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 [×3], C3, C3 [×3], C22, S3 [×7], C6 [×3], C9, C9 [×3], C32 [×2], C32, D6 [×3], D9 [×4], C18, C3×S3 [×5], C3⋊S3 [×2], C3×C9 [×2], C3×C9, He3, 3- 1+2 [×3], D18, S32 [×2], C3×D9 [×2], S3×C9 [×3], C32⋊C6 [×2], C9⋊C6 [×2], C9⋊S3 [×2], He3⋊C2, C9○He3, S3×D9 [×2], C32⋊D6, He3.4S3 [×2], He3.4C6, He3.6D6
Quotients: C1, C2 [×3], C22, S3 [×3], D6 [×3], S32 [×3], 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 26 17)(2 18 27)(3 10 19)(4 20 11)(5 12 21)(6 22 13)(7 14 23)(8 24 15)(9 16 25)
(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 22 16)(11 17 23)(12 24 18)(13 19 25)(14 26 20)(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 3)(4 9)(5 8)(6 7)(10 17)(11 16)(12 15)(13 14)(18 27)(19 26)(20 25)(21 24)(22 23)

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

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

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

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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