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G = He3⋊6D6  order 324 = 22·34

2nd semidirect product of He3 and D6 acting via D6/C3=C22

non-abelian, supersoluble, monomial

Aliases: He3⋊6D6, C33⋊6D6, C32⋊2S32, He3⋊C2⋊3S3, C33⋊C2⋊3S3, C3⋊2(C32⋊D6), (C3×He3)⋊5C22, He3⋊4S3⋊3C2, C3.2(C32⋊4D6), (C3×He3⋊C2)⋊3C2, SmallGroup(324,124)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C3×He3 — He3⋊6D6
Chief series
C1 — C3 — C32 — C33 — C3×He3 — C3×He3⋊C2 — He3⋊6D6
Lower central
C3×He3 — He3⋊6D6
Upper central
C1

Generators and relations for He3⋊6D6
 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, bd=db, ebe=b-1, dcd-1=c-1, ce=ec, ede=d-1 >

Subgroups: 1086 in 127 conjugacy classes, 19 normal (8 characteristic)
C1, C2, C3, C3, C3, C22, S3, C6, C32, C32, C32, D6, C3×S3, C3⋊S3, C3×C6, He3, He3, C33, C33, S32, C2×C3⋊S3, C32⋊C6, He3⋊C2, S3×C32, C3×C3⋊S3, C33⋊C2, C3×He3, C32⋊D6, S3×C3⋊S3, He3⋊4S3, C3×He3⋊C2, He3⋊6D6
Quotients: C1, C2, C22, S3, D6, S32, C32⋊D6, C32⋊4D6, He3⋊6D6

Character table of He3⋊6D6

 class 12A2B2C3A3B3C3D3E3F3G3H3I3J3K6A6B6C6D6E6F
 size 1927272222661212121212181818185454
ρ1111111111111111111111    trivial
ρ21-11-111111111111-1-1-1-1-11    linear of order 2
ρ31-1-1111111111111-1-1-1-11-1    linear of order 2
ρ411-1-1111111111111111-1-1    linear of order 2
ρ52-200-1-12-122-1-12-1-1-211100    orthogonal lifted from D6
ρ6202022222-1-1-1-1-1200000-1    orthogonal lifted from S3
ρ7200-22222-12-1-1-12-1000010    orthogonal lifted from D6
ρ820022222-12-1-1-12-10000-10    orthogonal lifted from S3
ρ920-2022222-1-1-1-1-12000001    orthogonal lifted from D6
ρ102200-1-12-122-1-12-1-12-1-1-100    orthogonal lifted from S3
ρ1140004444-2-2111-2-2000000    orthogonal lifted from S32
ρ124000-2-24-2-2411-2-21000000    orthogonal lifted from S32
ρ134000-2-24-24-211-21-2000000    orthogonal lifted from S32
ρ144000-2-24-2-2-2-1+3√-3/2-1-3√-3/2111000000    complex lifted from C32⋊4D6
ρ154000-2-24-2-2-2-1-3√-3/2-1+3√-3/2111000000    complex lifted from C32⋊4D6
ρ166-2006-3-3-3000000011-2100    orthogonal lifted from C32⋊D6
ρ176-200-36-3-30000000111-200    orthogonal lifted from C32⋊D6
ρ186-200-3-3-3600000001-21100    orthogonal lifted from C32⋊D6
ρ196200-36-3-30000000-1-1-1200    orthogonal lifted from C32⋊D6
ρ206200-3-3-360000000-12-1-100    orthogonal lifted from C32⋊D6
ρ2162006-3-3-30000000-1-12-100    orthogonal lifted from C32⋊D6

Permutation representations of He3⋊6D6
►On 27 points - transitive group 27T120
Generators in S27
(10 27 16)(11 17 22)(12 23 18)(13 19 24)(14 25 20)(15 21 26)

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

(1 23 26)(2 27 24)(3 25 22)(4 14 11)(5 12 15)(6 10 13)(7 20 17)(8 18 21)(9 16 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 3)(4 8)(5 7)(6 9)(10 16)(11 21)(12 20)(13 19)(14 18)(15 17)(22 26)(23 25)
 
G:=sub<Sym(27)| (10,27,16)(11,17,22)(12,23,18)(13,19,24)(14,25,20)(15,21,26), (1,5,8)(2,6,9)(3,4,7)(10,16,27)(11,17,22)(12,18,23)(13,19,24)(14,20,25)(15,21,26), (1,23,26)(2,27,24)(3,25,22)(4,14,11)(5,12,15)(6,10,13)(7,20,17)(8,18,21)(9,16,19), (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), (1,3)(4,8)(5,7)(6,9)(10,16)(11,21)(12,20)(13,19)(14,18)(15,17)(22,26)(23,25)>;
 
G:=Group( (10,27,16)(11,17,22)(12,23,18)(13,19,24)(14,25,20)(15,21,26), (1,5,8)(2,6,9)(3,4,7)(10,16,27)(11,17,22)(12,18,23)(13,19,24)(14,20,25)(15,21,26), (1,23,26)(2,27,24)(3,25,22)(4,14,11)(5,12,15)(6,10,13)(7,20,17)(8,18,21)(9,16,19), (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), (1,3)(4,8)(5,7)(6,9)(10,16)(11,21)(12,20)(13,19)(14,18)(15,17)(22,26)(23,25) );
 
G=PermutationGroup([[(10,27,16),(11,17,22),(12,23,18),(13,19,24),(14,25,20),(15,21,26)], [(1,5,8),(2,6,9),(3,4,7),(10,16,27),(11,17,22),(12,18,23),(13,19,24),(14,20,25),(15,21,26)], [(1,23,26),(2,27,24),(3,25,22),(4,14,11),(5,12,15),(6,10,13),(7,20,17),(8,18,21),(9,16,19)], [(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)], [(1,3),(4,8),(5,7),(6,9),(10,16),(11,21),(12,20),(13,19),(14,18),(15,17),(22,26),(23,25)]])
 
G:=TransitiveGroup(27,120);
 

Matrix representation of He3⋊6D6 ►in GL10(𝔽7)

6010000000
0601000000
6000000000
0600000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
1000000000
0100000000
0010000000
0001000000
0000610000
0000600000
0000006100
0000006000
0000000061
0000000060
,
6600000000
1000000000
0066000000
0010000000
0000100000
0000010000
0000000600
0000001600
0000000061
0000000060
,
3132000000
5464000000
6346000000
4123000000
0000060000
0000160000
0000000006
0000000016
0000000600
0000001600
,
5464000000
3132000000
4123000000
6346000000
0000610000
0000010000
0000000061
0000000001
0000006100
0000000100

G:=sub<GL(10,GF(7))| [6,0,6,0,0,0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0],[6,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0],[3,5,6,4,0,0,0,0,0,0,1,4,3,1,0,0,0,0,0,0,3,6,4,2,0,0,0,0,0,0,2,4,6,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0],[5,3,4,6,0,0,0,0,0,0,4,1,1,3,0,0,0,0,0,0,6,3,2,4,0,0,0,0,0,0,4,2,3,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,1,0,0] >;
 

He3⋊6D6 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_6D_6
 
% in TeX
 
G:=Group("He3:6D6");
 
// GroupNames label
 
G:=SmallGroup(324,124);
 
// by ID
 
G=gap.SmallGroup(324,124);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,80,297,2164,1096,3899]);
 
// 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,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⋊6D6 in TeX

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