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G = He3⋊D4order 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

C1C3He3He3⋊C2 — He3⋊D4
C1C3He3He3⋊C2C32⋊D6 — He3⋊D4
He3He3⋊C2 — He3⋊D4
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
2C3⋊S3
2C3⋊S3
6C3×S3
6C3×S3
6C3×S3
6C3×S3
9D12
6S32
6S32
2C32⋊C6
2C32⋊C6

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
ρ126200-3000-1003-3    orthogonal faithful
ρ136200-3000-100-33    orthogonal faithful

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

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

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

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

G:=TransitiveGroup(18,105);

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

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

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

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

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)

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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