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## G = He3⋊D4order 216 = 23·33

### The semidirect product of He3 and D4 acting faithfully

Aliases: He3⋊D4, C3.S3≀C2, He3⋊C4⋊C2, C32⋊D6⋊C2, He3⋊C2.2C22, SmallGroup(216,87)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C2 — He3⋊D4
 Chief series C1 — C3 — He3 — He3⋊C2 — C32⋊D6 — He3⋊D4
 Lower central He3 — He3⋊C2 — He3⋊D4
 Upper central C1

Generators and relations for He3⋊D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, eae=ab=ba, cac-1=dcd-1=ab-1, dad-1=bc-1, bc=cb, bd=db, ebe=b-1, ece=c-1, ede=d-1 >

9C2
18C2
18C2
6C3
6C3
9C4
27C22
27C22
6S3
6S3
6S3
6S3
9C6
18C6
18S3
18S3
18C6
2C32
2C32
27D4
9C12
9D6
9D6
18D6
18D6
9D12
6S32
6S32

Character table of He3⋊D4

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 12A 12B size 1 9 18 18 2 12 12 18 18 36 36 18 18 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 -2 0 0 2 2 2 0 -2 0 0 0 0 orthogonal lifted from D4 ρ6 4 0 -2 0 4 1 -2 0 0 1 0 0 0 orthogonal lifted from S3≀C2 ρ7 4 0 0 -2 4 -2 1 0 0 0 1 0 0 orthogonal lifted from S3≀C2 ρ8 4 0 2 0 4 1 -2 0 0 -1 0 0 0 orthogonal lifted from S3≀C2 ρ9 4 0 0 2 4 -2 1 0 0 0 -1 0 0 orthogonal lifted from S3≀C2 ρ10 6 -2 0 0 -3 0 0 2 1 0 0 -1 -1 orthogonal faithful ρ11 6 -2 0 0 -3 0 0 -2 1 0 0 1 1 orthogonal faithful ρ12 6 2 0 0 -3 0 0 0 -1 0 0 √3 -√3 orthogonal faithful ρ13 6 2 0 0 -3 0 0 0 -1 0 0 -√3 √3 orthogonal faithful

Permutation representations of He3⋊D4
On 18 points - transitive group 18T105
Generators in S18
(1 18 14)(2 13 7)(3 8 16)(4 15 11)(5 9 17)(6 12 10)
(1 6 3)(2 5 4)(7 17 11)(8 18 12)(9 15 13)(10 16 14)
(1 16 18)(2 9 7)(3 10 8)(4 13 11)(5 15 17)(6 14 12)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 5)(2 6)(3 4)(7 14)(8 13)(9 12)(10 11)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(18,105);

On 27 points - transitive group 27T76
Generators in S27
(1 22 14)(2 11 20)(3 12 9)(4 13 18)(5 10 6)(7 24 23)(8 27 19)(15 26 25)(16 21 17)
(1 3 2)(4 19 24)(5 16 25)(6 17 26)(7 18 27)(8 23 13)(9 20 14)(10 21 15)(11 22 12)
(1 8 10)(2 13 15)(3 23 21)(4 16 14)(5 20 24)(6 12 18)(7 26 22)(9 19 25)(11 27 17)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
(2 3)(4 5)(6 7)(8 10)(12 22)(13 21)(14 20)(15 23)(16 24)(17 27)(18 26)(19 25)

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

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

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

G:=TransitiveGroup(27,76);

He3⋊D4 is a maximal subgroup of   He3⋊SD16
He3⋊D4 is a maximal quotient of   He32SD16  He3⋊D8  He3⋊Q16  C6.S3≀C2  C32⋊D6⋊C4

Matrix representation of He3⋊D4 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 1 0 -1 0 -1 1 -1 0 -1 -2 -2 -1 0 -1 1 1 1 0 0 0 0 1 1 0
,
 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 1 0 1 0 0 1 0 -1 0 -1 -1 -1
,
 1 -1 -1 1 -1 1 0 -1 -2 -1 -2 -1 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 2 0 1 0 0 1 1 0 1 0
,
 0 0 1 0 0 0 0 0 0 1 0 0 -1 -1 0 -1 -1 -2 0 0 0 1 1 -1 0 1 0 -1 0 1 0 0 0 0 0 1
,
 -1 -1 0 -2 -2 -1 0 0 0 -1 -1 1 0 0 1 -1 0 0 0 0 0 -1 0 0 0 0 0 2 1 0 0 1 0 1 1 0

G:=sub<GL(6,Integers())| [1,0,1,-1,0,0,0,1,0,0,-1,0,0,0,-1,-1,1,0,0,0,0,-2,1,1,0,0,-1,-2,1,1,0,0,1,-1,0,0],[-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,-1,1,0,0,0,1,0,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,-1,-1,0,0,0,1,-1,-2,-1,-1,2,1,1,-1,1,0,0,0,-1,-2,0,0,1,1,1,-1,0,0,0,0],[0,0,-1,0,0,0,0,0,-1,0,1,0,1,0,0,0,0,0,0,1,-1,1,-1,0,0,0,-1,1,0,0,0,0,-2,-1,1,1],[-1,0,0,0,0,0,-1,0,0,0,0,1,0,0,1,0,0,0,-2,-1,-1,-1,2,1,-2,-1,0,0,1,1,-1,1,0,0,0,0] >;

He3⋊D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes D_4
% in TeX

G:=Group("He3:D4");
// GroupNames label

G:=SmallGroup(216,87);
// by ID

G=gap.SmallGroup(216,87);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,73,387,297,111,130,376,382,5189,2603]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,e*a*e=a*b=b*a,c*a*c^-1=d*c*d^-1=a*b^-1,d*a*d^-1=b*c^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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