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## G = He3⋊5D6order 324 = 22·34

### 1st semidirect product of He3 and D6 acting via D6/C3=C22

Aliases: He35D6, C334D6, C321S32, C32⋊C6⋊S3, C33⋊C22S3, C31(C32⋊D6), (C3×He3)⋊3C22, He34S32C2, He35S32C2, C3⋊S3⋊(C3⋊S3), C32⋊(C2×C3⋊S3), (C3×C3⋊S3)⋊2S3, C3.3(S3×C3⋊S3), (C3×C32⋊C6)⋊3C2, SmallGroup(324,121)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — He3⋊5D6
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C3×C32⋊C6 — He3⋊5D6
 Lower central C3×He3 — He3⋊5D6
 Upper central C1

Generators and relations for He35D6
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, dbd-1=b-1, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1122 in 133 conjugacy classes, 24 normal (15 characteristic)
C1, C2 [×3], C3 [×2], C3 [×9], C22, S3 [×17], C6 [×6], C32 [×2], C32 [×3], C32 [×10], D6 [×6], C3×S3 [×17], C3⋊S3, C3⋊S3 [×9], C3×C6, He3 [×3], He3 [×3], C33 [×2], C33, S32 [×6], C2×C3⋊S3, C32⋊C6 [×3], C32⋊C6 [×3], He3⋊C2 [×6], S3×C32, C3×C3⋊S3, C3×C3⋊S3 [×4], C33⋊C2, C3×He3, C32⋊D6 [×3], S3×C3⋊S3, C324D6, C3×C32⋊C6, He34S3, He35S3, He35D6
Quotients: C1, C2 [×3], C22, S3 [×5], D6 [×5], C3⋊S3, S32 [×4], C2×C3⋊S3, C32⋊D6, S3×C3⋊S3, He35D6

Character table of He35D6

 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 4 6 6 6 6 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 2 -1 -1 -1 -1 2 2 -1 -1 2 -1 1 -2 1 1 0 0 orthogonal lifted from D6 ρ6 2 2 0 0 2 2 2 -1 -1 -1 2 -1 2 -1 -1 -1 -1 -1 2 0 0 orthogonal lifted from S3 ρ7 2 0 0 -2 2 2 2 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 1 0 orthogonal lifted from D6 ρ8 2 2 0 0 2 -1 -1 2 -1 -1 2 -1 -1 -1 2 -1 -1 2 -1 0 0 orthogonal lifted from S3 ρ9 2 0 0 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 -1 0 orthogonal lifted from S3 ρ10 2 2 0 0 2 -1 -1 -1 2 -1 2 2 -1 -1 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ11 2 2 0 0 2 -1 -1 -1 -1 2 2 -1 -1 2 -1 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ12 2 -2 0 0 2 -1 -1 2 -1 -1 2 -1 -1 -1 2 1 1 -2 1 0 0 orthogonal lifted from D6 ρ13 2 -2 0 0 2 2 2 -1 -1 -1 2 -1 2 -1 -1 1 1 1 -2 0 0 orthogonal lifted from D6 ρ14 2 -2 0 0 2 -1 -1 -1 2 -1 2 2 -1 -1 -1 -2 1 1 1 0 0 orthogonal lifted from D6 ρ15 4 0 0 0 4 -2 -2 -2 -2 4 -2 1 1 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ16 4 0 0 0 4 -2 -2 4 -2 -2 -2 1 1 1 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ17 4 0 0 0 4 4 4 -2 -2 -2 -2 1 -2 1 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ18 4 0 0 0 4 -2 -2 -2 4 -2 -2 -2 1 1 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ19 6 0 -2 0 -3 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 orthogonal lifted from C32⋊D6 ρ20 6 0 2 0 -3 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 orthogonal lifted from C32⋊D6 ρ21 12 0 0 0 -6 -6 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of He35D6
On 18 points - transitive group 18T133
Generators in S18
(1 11 15)(2 16 12)(3 7 17)(4 18 8)(5 9 13)(6 14 10)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)(13 17 15)(14 16 18)
(7 11 9)(8 12 10)(13 15 17)(14 16 18)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 16)(14 15)(17 18)

G:=sub<Sym(18)| (1,11,15)(2,16,12)(3,7,17)(4,18,8)(5,9,13)(6,14,10), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (7,11,9)(8,12,10)(13,15,17)(14,16,18), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18)>;

G:=Group( (1,11,15)(2,16,12)(3,7,17)(4,18,8)(5,9,13)(6,14,10), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (7,11,9)(8,12,10)(13,15,17)(14,16,18), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18) );

G=PermutationGroup([(1,11,15),(2,16,12),(3,7,17),(4,18,8),(5,9,13),(6,14,10)], [(1,5,3),(2,4,6),(7,11,9),(8,10,12),(13,17,15),(14,16,18)], [(7,11,9),(8,12,10),(13,15,17),(14,16,18)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,16),(14,15),(17,18)])

G:=TransitiveGroup(18,133);

On 18 points - transitive group 18T139
Generators in S18
(7 9 11)(8 12 10)(13 17 15)(14 16 18)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)(13 17 15)(14 16 18)
(1 11 15)(2 12 16)(3 7 17)(4 8 18)(5 9 13)(6 10 14)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 6)(2 5)(3 4)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)

G:=sub<Sym(18)| (7,9,11)(8,12,10)(13,17,15)(14,16,18), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (1,11,15)(2,12,16)(3,7,17)(4,8,18)(5,9,13)(6,10,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)>;

G:=Group( (7,9,11)(8,12,10)(13,17,15)(14,16,18), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (1,11,15)(2,12,16)(3,7,17)(4,8,18)(5,9,13)(6,10,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13) );

G=PermutationGroup([(7,9,11),(8,12,10),(13,17,15),(14,16,18)], [(1,5,3),(2,4,6),(7,11,9),(8,10,12),(13,17,15),(14,16,18)], [(1,11,15),(2,12,16),(3,7,17),(4,8,18),(5,9,13),(6,10,14)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)])

G:=TransitiveGroup(18,139);

On 27 points - transitive group 27T117
Generators in S27
(1 10 13)(2 14 11)(3 12 15)(4 24 17)(5 18 25)(6 26 19)(7 20 27)(8 22 21)(9 16 23)
(1 5 8)(2 9 6)(3 7 4)(10 18 22)(11 23 19)(12 20 24)(13 25 21)(14 16 26)(15 27 17)
(1 3 2)(4 6 8)(5 7 9)(10 20 26)(11 21 27)(12 16 22)(13 17 23)(14 18 24)(15 19 25)
(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 2)(5 9)(6 8)(10 11)(12 15)(13 14)(16 25)(17 24)(18 23)(19 22)(20 27)(21 26)

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

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

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

G:=TransitiveGroup(27,117);

On 27 points - transitive group 27T127
Generators in S27
(1 16 19)(2 20 17)(3 18 21)(4 11 14)(5 15 12)(6 13 10)
(1 19 16)(2 17 20)(3 21 18)(4 11 14)(5 15 12)(6 13 10)(7 26 23)(8 24 27)(9 22 25)
(1 5 9)(2 6 7)(3 4 8)(10 23 20)(11 24 21)(12 25 16)(13 26 17)(14 27 18)(15 22 19)
(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)(3 6)(7 8)(10 18)(11 17)(12 16)(13 21)(14 20)(15 19)(23 27)(24 26)

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

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

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

G:=TransitiveGroup(27,127);

Matrix representation of He35D6 in GL10(ℤ)

 -1 0 -1 0 0 0 0 0 0 0 0 -1 0 -1 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 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 1 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 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 -1
,
 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 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 1 -1 0 0 0 0 0 0 1 0 0 0 -1
,
 -1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 -1 1 -1 0 0 0 0 0 0 1 0 1 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 -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 -1 0 0 0 0 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 -1 1 -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 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0

G:=sub<GL(10,Integers())| [-1,0,1,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,0,0,0,-1],[-1,-1,1,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,0,-1,0],[-1,-1,1,1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0] >;

He35D6 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5D_6
% in TeX

G:=Group("He3:5D6");
// GroupNames label

G:=SmallGroup(324,121);
// by ID

G=gap.SmallGroup(324,121);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,297,2164,1096,7781,3899]);
// 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,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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