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## G = He3⋊6D6order 324 = 22·34

### 2nd semidirect product of He3 and D6 acting via D6/C3=C22

Aliases: He36D6, C336D6, C322S32, He3⋊C23S3, C33⋊C23S3, C32(C32⋊D6), (C3×He3)⋊5C22, He34S33C2, C3.2(C324D6), (C3×He3⋊C2)⋊3C2, SmallGroup(324,124)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — He3⋊6D6
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C3×He3⋊C2 — He3⋊6D6
 Lower central C3×He3 — He3⋊6D6
 Upper central C1

Generators and relations for He36D6
G = < a,b,c,d,e | a3=b3=c3=d6=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=c-1, ce=ec, ede=d-1 >

Subgroups: 1086 in 127 conjugacy classes, 19 normal (8 characteristic)
C1, C2 [×3], C3, C3 [×3], C3 [×6], C22, S3 [×15], C6 [×6], C32, C32 [×2], C32 [×13], D6 [×6], C3×S3 [×20], C3⋊S3 [×12], C3×C6, He3, He3 [×3], C33 [×2], C33, S32 [×8], C2×C3⋊S3, C32⋊C6 [×4], He3⋊C2, S3×C32 [×3], C3×C3⋊S3 [×2], C33⋊C2 [×2], C3×He3, C32⋊D6, S3×C3⋊S3 [×2], He34S3 [×2], C3×He3⋊C2, He36D6
Quotients: C1, C2 [×3], C22, S3 [×3], D6 [×3], S32 [×3], C32⋊D6 [×3], C324D6, He36D6

Character table of He36D6

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 6A 6B 6C 6D 6E 6F size 1 9 27 27 2 2 2 2 6 6 12 12 12 12 12 18 18 18 18 54 54 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 -1 -1 2 -1 2 2 -1 -1 2 -1 -1 -2 1 1 1 0 0 orthogonal lifted from D6 ρ6 2 0 2 0 2 2 2 2 2 -1 -1 -1 -1 -1 2 0 0 0 0 0 -1 orthogonal lifted from S3 ρ7 2 0 0 -2 2 2 2 2 -1 2 -1 -1 -1 2 -1 0 0 0 0 1 0 orthogonal lifted from D6 ρ8 2 0 0 2 2 2 2 2 -1 2 -1 -1 -1 2 -1 0 0 0 0 -1 0 orthogonal lifted from S3 ρ9 2 0 -2 0 2 2 2 2 2 -1 -1 -1 -1 -1 2 0 0 0 0 0 1 orthogonal lifted from D6 ρ10 2 2 0 0 -1 -1 2 -1 2 2 -1 -1 2 -1 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ11 4 0 0 0 4 4 4 4 -2 -2 1 1 1 -2 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ12 4 0 0 0 -2 -2 4 -2 -2 4 1 1 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ13 4 0 0 0 -2 -2 4 -2 4 -2 1 1 -2 1 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ14 4 0 0 0 -2 -2 4 -2 -2 -2 -1+3√-3/2 -1-3√-3/2 1 1 1 0 0 0 0 0 0 complex lifted from C32⋊4D6 ρ15 4 0 0 0 -2 -2 4 -2 -2 -2 -1-3√-3/2 -1+3√-3/2 1 1 1 0 0 0 0 0 0 complex lifted from C32⋊4D6 ρ16 6 -2 0 0 6 -3 -3 -3 0 0 0 0 0 0 0 1 1 -2 1 0 0 orthogonal lifted from C32⋊D6 ρ17 6 -2 0 0 -3 6 -3 -3 0 0 0 0 0 0 0 1 1 1 -2 0 0 orthogonal lifted from C32⋊D6 ρ18 6 -2 0 0 -3 -3 -3 6 0 0 0 0 0 0 0 1 -2 1 1 0 0 orthogonal lifted from C32⋊D6 ρ19 6 2 0 0 -3 6 -3 -3 0 0 0 0 0 0 0 -1 -1 -1 2 0 0 orthogonal lifted from C32⋊D6 ρ20 6 2 0 0 -3 -3 -3 6 0 0 0 0 0 0 0 -1 2 -1 -1 0 0 orthogonal lifted from C32⋊D6 ρ21 6 2 0 0 6 -3 -3 -3 0 0 0 0 0 0 0 -1 -1 2 -1 0 0 orthogonal lifted from C32⋊D6

Permutation representations of He36D6
On 27 points - transitive group 27T120
Generators in S27
(10 25 19)(11 20 26)(12 27 21)(13 16 22)(14 23 17)(15 18 24)
(1 5 8)(2 6 9)(3 4 7)(10 19 25)(11 20 26)(12 21 27)(13 16 22)(14 17 23)(15 18 24)
(1 23 26)(2 27 24)(3 25 22)(4 10 13)(5 14 11)(6 12 15)(7 19 16)(8 17 20)(9 21 18)
(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 8)(5 7)(6 9)(10 17)(11 16)(12 21)(13 20)(14 19)(15 18)(22 26)(23 25)

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

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

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

G:=TransitiveGroup(27,120);

Matrix representation of He36D6 in GL10(𝔽7)

 6 0 1 0 0 0 0 0 0 0 0 6 0 1 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 6 0
,
 6 6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 6 6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 6 0
,
 3 1 3 2 0 0 0 0 0 0 5 4 6 4 0 0 0 0 0 0 6 3 4 6 0 0 0 0 0 0 4 1 2 3 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 1 6 0 0
,
 5 4 6 4 0 0 0 0 0 0 3 1 3 2 0 0 0 0 0 0 4 1 2 3 0 0 0 0 0 0 6 3 4 6 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(10,GF(7))| [6,0,6,0,0,0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0],[6,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0],[3,5,6,4,0,0,0,0,0,0,1,4,3,1,0,0,0,0,0,0,3,6,4,2,0,0,0,0,0,0,2,4,6,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0],[5,3,4,6,0,0,0,0,0,0,4,1,1,3,0,0,0,0,0,0,6,3,2,4,0,0,0,0,0,0,4,2,3,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,1,0,0] >;

He36D6 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_6D_6
% in TeX

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

G:=SmallGroup(324,124);
// by ID

G=gap.SmallGroup(324,124);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,80,297,2164,1096,3899]);
// Polycyclic

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

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