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G = He3⋊2D4  order 216 = 23·33

1st semidirect product of He3 and D4 acting via D4/C2=C22

non-abelian, supersoluble, monomial

Aliases: He3⋊2D4, C6.18S32, (C3×C6).3D6, He3⋊3C4⋊1C2, C32⋊1(C3⋊D4), C2.4(C32⋊D6), C3.1(D6⋊S3), (C2×He3).3C22, (C2×C3⋊S3)⋊1S3, (C2×C32⋊C6)⋊1C2, SmallGroup(216,35)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×He3 — He3⋊2D4
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — He3⋊2D4
Lower central
He3 — C2×He3 — He3⋊2D4
Upper central
C1 — C2

Generators and relations for He3⋊2D4
 G = < a,b,c,d,e | a3=b3=c3=d4=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: 352 in 62 conjugacy classes, 16 normal (8 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C3⋊D4, He3, C3×Dic3, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, C3⋊D12, He3⋊3C4, C2×C32⋊C6, He3⋊2D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S32, D6⋊S3, C32⋊D6, He3⋊2D4

Character table of He3⋊2D4

 class 12A2B2C3A3B3C3D46A6B6C6D6E6F6G6H12A12B
 size 111818266121826612181818181818
ρ11111111111111111111    trivial
ρ211-1-1111111111-1-1-1-111    linear of order 2
ρ3111-11111-11111-111-1-1-1    linear of order 2
ρ411-111111-111111-1-11-1-1    linear of order 2
ρ52-20022220-2-2-2-2000000    orthogonal lifted from D4
ρ6222022-1-1022-1-10-1-1000    orthogonal lifted from S3
ρ722022-12-102-12-1-100-100    orthogonal lifted from S3
ρ822-2022-1-1022-1-1011000    orthogonal lifted from D6
ρ9220-22-12-102-12-1100100    orthogonal lifted from D6
ρ102-20022-1-10-2-2110√-3-√-3000    complex lifted from C3⋊D4
ρ112-2002-12-10-21-21-√-300√-300    complex lifted from C3⋊D4
ρ122-20022-1-10-2-2110-√-3√-3000    complex lifted from C3⋊D4
ρ132-2002-12-10-21-21√-300-√-300    complex lifted from C3⋊D4
ρ1444004-2-2104-2-21000000    orthogonal lifted from S32
ρ154-4004-2-210-422-1000000    symplectic lifted from D6⋊S3, Schur index 2
ρ166600-3000-2-3000000011    orthogonal lifted from C32⋊D6
ρ176600-30002-30000000-1-1    orthogonal lifted from C32⋊D6
ρ186-600-3000030000000√3-√3    orthogonal faithful
ρ196-600-3000030000000-√3√3    orthogonal faithful

Smallest permutation representation of He3⋊2D4
►On 36 points
Generators in S36
(1 30 13)(2 14 31)(3 32 15)(4 16 29)(5 18 9)(6 10 19)(7 20 11)(8 12 17)(21 34 28)(22 25 35)(23 36 26)(24 27 33)

(1 5 24)(2 6 21)(3 7 22)(4 8 23)(9 33 13)(10 34 14)(11 35 15)(12 36 16)(17 26 29)(18 27 30)(19 28 31)(20 25 32)

(9 13 33)(10 34 14)(11 15 35)(12 36 16)(17 29 26)(18 27 30)(19 31 28)(20 25 32)

(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 28)(29 30 31 32)(33 34 35 36)

(1 4)(2 3)(5 23)(6 22)(7 21)(8 24)(9 36)(10 35)(11 34)(12 33)(13 16)(14 15)(17 27)(18 26)(19 25)(20 28)(29 30)(31 32)
 
G:=sub<Sym(36)| (1,30,13)(2,14,31)(3,32,15)(4,16,29)(5,18,9)(6,10,19)(7,20,11)(8,12,17)(21,34,28)(22,25,35)(23,36,26)(24,27,33), (1,5,24)(2,6,21)(3,7,22)(4,8,23)(9,33,13)(10,34,14)(11,35,15)(12,36,16)(17,26,29)(18,27,30)(19,28,31)(20,25,32), (9,13,33)(10,34,14)(11,15,35)(12,36,16)(17,29,26)(18,27,30)(19,31,28)(20,25,32), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,4)(2,3)(5,23)(6,22)(7,21)(8,24)(9,36)(10,35)(11,34)(12,33)(13,16)(14,15)(17,27)(18,26)(19,25)(20,28)(29,30)(31,32)>;
 
G:=Group( (1,30,13)(2,14,31)(3,32,15)(4,16,29)(5,18,9)(6,10,19)(7,20,11)(8,12,17)(21,34,28)(22,25,35)(23,36,26)(24,27,33), (1,5,24)(2,6,21)(3,7,22)(4,8,23)(9,33,13)(10,34,14)(11,35,15)(12,36,16)(17,26,29)(18,27,30)(19,28,31)(20,25,32), (9,13,33)(10,34,14)(11,15,35)(12,36,16)(17,29,26)(18,27,30)(19,31,28)(20,25,32), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,4)(2,3)(5,23)(6,22)(7,21)(8,24)(9,36)(10,35)(11,34)(12,33)(13,16)(14,15)(17,27)(18,26)(19,25)(20,28)(29,30)(31,32) );
 
G=PermutationGroup([[(1,30,13),(2,14,31),(3,32,15),(4,16,29),(5,18,9),(6,10,19),(7,20,11),(8,12,17),(21,34,28),(22,25,35),(23,36,26),(24,27,33)], [(1,5,24),(2,6,21),(3,7,22),(4,8,23),(9,33,13),(10,34,14),(11,35,15),(12,36,16),(17,26,29),(18,27,30),(19,28,31),(20,25,32)], [(9,13,33),(10,34,14),(11,15,35),(12,36,16),(17,29,26),(18,27,30),(19,31,28),(20,25,32)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,4),(2,3),(5,23),(6,22),(7,21),(8,24),(9,36),(10,35),(11,34),(12,33),(13,16),(14,15),(17,27),(18,26),(19,25),(20,28),(29,30),(31,32)]])
 

He3⋊2D4 is a maximal subgroup of   He3⋊2SD16  He3⋊D8  C12⋊S3⋊S3  C12.91S32  C12.86S32  C62.9D6  C62⋊D6
He3⋊2D4 is a maximal quotient of   He3⋊3SD16  He3⋊2D8  He3⋊2Q16  C62.3D6  C62.4D6

Matrix representation of He3⋊2D4 ►in GL6(𝔽13)

0000121
33001112
000040
100040
0010100
0001100
,
0120000
1120000
90121200
041000
30001212
0100010
,
100000
010000
99121200
001000
000001
33001212
,
370000
6100000
10800107
530063
10510700
836300
,
730000
1060000
350036
12100310
383600
4931000

G:=sub<GL(6,GF(13))| [0,3,0,1,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,11,4,4,10,10,1,12,0,0,0,0],[0,1,9,0,3,0,12,12,0,4,0,10,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,9,0,0,3,0,1,9,0,0,3,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[3,6,10,5,10,8,7,10,8,3,5,3,0,0,0,0,10,6,0,0,0,0,7,3,0,0,10,6,0,0,0,0,7,3,0,0],[7,10,3,12,3,4,3,6,5,1,8,9,0,0,0,0,3,3,0,0,0,0,6,10,0,0,3,3,0,0,0,0,6,10,0,0] >;
 

He3⋊2D4 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_2D_4
 
% in TeX
 
G:=Group("He3:2D4");
 
// GroupNames label
 
G:=SmallGroup(216,35);
 
// by ID
 
G=gap.SmallGroup(216,35);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,201,1444,382,5189,2603]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=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
 

Export

Character table of He3⋊2D4 in TeX

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