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G = He3⋊D4  order 216 = 23·33

The semidirect product of He3 and D4 acting faithfully

non-abelian, soluble

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 >

C1
9C2
18C2
18C2
C3
6C3
6C3
9C4
27C22
27C22
6S3
6S3
6S3
6S3
9C6
18C6
18S3
18S3
18C6
2C32
2C32
27D4
9C12
9D6
9D6
18D6
18D6
2C3⋊S3
2C3⋊S3
6C3×S3
6C3×S3
6C3×S3
6C3×S3
He3
9D12
6S32
6S32
He3⋊C2
2C32⋊C6
2C32⋊C6
C32⋊D6
He3⋊C4
C32⋊D6
He3⋊D4

Character table of He3⋊D4

 class 12A2B2C3A3B3C46A6B6C12A12B
 size 19181821212181836361818
ρ11111111111111    trivial
ρ211-11111-11-11-1-1    linear of order 2
ρ3111-1111-111-1-1-1    linear of order 2
ρ411-1-111111-1-111    linear of order 2
ρ52-2002220-20000    orthogonal lifted from D4
ρ640-2041-2001000    orthogonal lifted from S3≀C2
ρ7400-24-21000100    orthogonal lifted from S3≀C2
ρ8402041-200-1000    orthogonal lifted from S3≀C2
ρ940024-21000-100    orthogonal lifted from S3≀C2
ρ106-200-3002100-1-1    orthogonal faithful
ρ116-200-300-210011    orthogonal faithful
ρ126200-3000-100√3-√3    orthogonal faithful
ρ136200-3000-100-√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   He3⋊2SD16  He3⋊D8  He3⋊Q16  C6.S3≀C2  C32⋊D6⋊C4

Matrix representation of He3⋊D4 ►in GL6(ℤ)

100000
010000
10-10-11
-10-1-2-2-1
0-11110
000110
,
-110000
-100000
00-1100
00-1000
101001
0-10-1-1-1
,
1-1-11-11
0-1-2-1-2-1
00-1100
00-1000
002010
011010
,
001000
000100
-1-10-1-1-2
00011-1
010-101
000001
,
-1-10-2-2-1
000-1-11
001-100
000-100
000210
010110

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
 

Export

Subgroup lattice of He3⋊D4 in TeX
Character table of He3⋊D4 in TeX

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