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G = He3.2D6order 324 = 22·34

2nd non-split extension by He3 of D6 acting faithfully

Aliases: He3.2D6, C9⋊S33S3, (C3×C9)⋊3D6, C32.3S32, He3⋊S3⋊C2, He3⋊C22S3, He3.2S3⋊C2, He3.2C6⋊C2, He3⋊C3⋊C22, C3.4(C32⋊D6), SmallGroup(324,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3⋊C3 — He3.2D6
 Chief series C1 — C3 — C32 — He3 — He3⋊C3 — He3.2C6 — He3.2D6
 Lower central He3⋊C3 — He3.2D6
 Upper central C1

Generators and relations for He3.2D6
G = < a,b,c,d,e | a3=b3=c3=e2=1, d6=b, ab=ba, cac-1=eae=ab-1, dad-1=a-1b, bc=cb, bd=db, ebe=b-1, dcd-1=ece=a-1c-1, ede=b-1d5 >

Subgroups: 596 in 56 conjugacy classes, 12 normal (all characteristic)
C1, C2 [×3], C3, C3 [×3], C22, S3 [×7], C6 [×3], C9, C32, C32 [×2], D6 [×3], D9 [×2], C18, C3×S3 [×4], C3⋊S3 [×3], C3×C9, He3, He3, D18, S32 [×2], C3×D9, S3×C9, C32⋊C6 [×3], C9⋊S3, He3⋊C2, He3⋊C3, S3×D9, C32⋊D6, He3.2C6, He3.2S3, He3⋊S3, He3.2D6
Quotients: C1, C2 [×3], C22, S3 [×2], D6 [×2], S32, C32⋊D6, He3.2D6

Character table of He3.2D6

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

Permutation representations of He3.2D6
On 27 points - transitive group 27T123
Generators in S27
(1 15 18)(2 25 10)(3 17 20)(4 27 12)(5 19 22)(6 11 14)(7 21 24)(8 13 16)(9 23 26)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(1 15 24)(2 10 19)(4 27 18)(5 22 13)(7 21 12)(8 16 25)(11 17 23)(14 26 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 7)(2 6)(3 5)(8 9)(10 20)(11 19)(12 18)(13 17)(14 16)(21 27)(22 26)(23 25)

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

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

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

G:=TransitiveGroup(27,123);

On 27 points - transitive group 27T133
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(10 16 22)(12 18 24)(14 20 26)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(1 16 25)(2 14 23)(3 12 21)(4 10 19)(5 26 17)(6 24 15)(7 22 13)(8 20 11)(9 18 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 5)(2 4)(6 9)(7 8)(10 11)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)

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

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

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

G:=TransitiveGroup(27,133);

Matrix representation of He3.2D6 in GL6(𝔽19)

 0 18 0 0 0 0 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 1 0 0 0 0 18 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 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
,
 7 14 0 0 0 0 5 2 0 0 0 0 0 0 0 0 14 17 0 0 0 0 2 12 0 0 14 17 0 0 0 0 2 12 0 0
,
 12 5 0 0 0 0 17 7 0 0 0 0 0 0 0 0 5 2 0 0 0 0 7 14 0 0 5 2 0 0 0 0 7 14 0 0

G:=sub<GL(6,GF(19))| [0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[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],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[7,5,0,0,0,0,14,2,0,0,0,0,0,0,0,0,14,2,0,0,0,0,17,12,0,0,14,2,0,0,0,0,17,12,0,0],[12,17,0,0,0,0,5,7,0,0,0,0,0,0,0,0,5,7,0,0,0,0,2,14,0,0,5,7,0,0,0,0,2,14,0,0] >;

He3.2D6 in GAP, Magma, Sage, TeX

{\rm He}_3._2D_6
% in TeX

G:=Group("He3.2D6");
// GroupNames label

G:=SmallGroup(324,41);
// by ID

G=gap.SmallGroup(324,41);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2024,500,579,303,5404,1090,382,3899]);
// Polycyclic

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

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