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G = He3.6D6  order 324 = 22·34

The non-split extension by He3 of D6 acting via D6/C3=C22

non-abelian, supersoluble, monomial

Aliases: He3.6D6, C9.2S32, C9⋊S3⋊4S3, (C3×C9)⋊4D6, C32.4S32, C9○He3⋊C22, He3.4C6⋊C2, He3.4S3⋊C2, He3⋊C2.3S3, C3.3(C32⋊4D6), SmallGroup(324,125)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C9○He3 — He3.6D6
Chief series
C1 — C3 — C32 — C3×C9 — C9○He3 — He3.4C6 — He3.6D6
Lower central
C9○He3 — He3.6D6
Upper central
C1

Generators and relations for He3.6D6
 G = < a,b,c,d,e | a3=b3=c3=e2=1, d6=ebe=b-1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, bd=db, dcd-1=c-1, ce=ec, ede=bd5 >

Subgroups: 555 in 71 conjugacy classes, 16 normal (8 characteristic)
C1, C2, C3, C3, C22, S3, C6, C9, C9, C32, C32, D6, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, 3- 1+2, D18, S32, C3×D9, S3×C9, C32⋊C6, C9⋊C6, C9⋊S3, He3⋊C2, C9○He3, S3×D9, C32⋊D6, He3.4S3, He3.4C6, He3.6D6
Quotients: C1, C2, C22, S3, D6, S32, C32⋊4D6, He3.6D6

Character table of He3.6D6

 class 12A2B2C3A3B3C3D6A6B6C9A9B9C9D9E9F9G18A18B18C
 size 1927272661218545422212121212181818
ρ1111111111111111111111    trivial
ρ21-11-11111-11-11111111-1-1-1    linear of order 2
ρ311-1-111111-1-11111111111    linear of order 2
ρ41-1-111111-1-111111111-1-1-1    linear of order 2
ρ52-2002222-200-1-1-1-1-1-1-1111    orthogonal lifted from D6
ρ6200-22-12-1001222-1-12-1000    orthogonal lifted from D6
ρ720-2022-1-1010222-12-1-1000    orthogonal lifted from D6
ρ820022-12-100-1222-1-12-1000    orthogonal lifted from S3
ρ922002222200-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ10202022-1-10-10222-12-1-1000    orthogonal lifted from S3
ρ11400044-2-2000-2-2-21-211000    orthogonal lifted from S32
ρ1240004-24-2000-2-2-211-21000    orthogonal lifted from S32
ρ1340004-2-210004441-2-21000    orthogonal lifted from S32
ρ1440004-2-21000-2-2-2-1+3√-3/211-1-3√-3/2000    complex lifted from C32⋊4D6
ρ1540004-2-21000-2-2-2-1-3√-3/211-1+3√-3/2000    complex lifted from C32⋊4D6
ρ166200-3000-1003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal faithful
ρ176200-3000-1003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal faithful
ρ186-200-30001003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000-ζ95-ζ94-ζ98-ζ9-ζ97-ζ92    orthogonal faithful
ρ196-200-30001003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000-ζ98-ζ9-ζ97-ζ92-ζ95-ζ94    orthogonal faithful
ρ206200-3000-1003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal faithful
ρ216-200-30001003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000-ζ97-ζ92-ζ95-ζ94-ζ98-ζ9    orthogonal faithful

Permutation representations of He3.6D6
►On 27 points - transitive group 27T122
Generators in S27
(1 23 14)(2 15 24)(3 25 16)(4 17 26)(5 27 18)(6 19 10)(7 11 20)(8 21 12)(9 13 22)

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

(10 16 22)(11 23 17)(12 18 24)(13 25 19)(14 20 26)(15 27 21)

(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 9)(5 8)(6 7)(10 11)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)
 
G:=sub<Sym(27)| (1,23,14)(2,15,24)(3,25,16)(4,17,26)(5,27,18)(6,19,10)(7,11,20)(8,21,12)(9,13,22), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (10,16,22)(11,23,17)(12,18,24)(13,25,19)(14,20,26)(15,27,21), (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)(4,9)(5,8)(6,7)(10,11)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)>;
 
G:=Group( (1,23,14)(2,15,24)(3,25,16)(4,17,26)(5,27,18)(6,19,10)(7,11,20)(8,21,12)(9,13,22), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (10,16,22)(11,23,17)(12,18,24)(13,25,19)(14,20,26)(15,27,21), (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)(4,9)(5,8)(6,7)(10,11)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20) );
 
G=PermutationGroup([[(1,23,14),(2,15,24),(3,25,16),(4,17,26),(5,27,18),(6,19,10),(7,11,20),(8,21,12),(9,13,22)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21)], [(10,16,22),(11,23,17),(12,18,24),(13,25,19),(14,20,26),(15,27,21)], [(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),(4,9),(5,8),(6,7),(10,11),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,20)]])
 
G:=TransitiveGroup(27,122);
 

Matrix representation of He3.6D6 ►in GL6(𝔽19)

100000
010000
0018100
0018000
18180181818
001010
,
1810000
1800000
0018100
0018000
101001
0180181818
,
001000
000100
0000181
181818181718
000010
100010
,
1220000
17140000
17171717310
5555123
2020142
20142142
,
1810000
010000
0018100
000100
101011
018018018

G:=sub<GL(6,GF(19))| [1,0,0,0,18,0,0,1,0,0,18,0,0,0,18,18,0,1,0,0,1,0,18,0,0,0,0,0,18,1,0,0,0,0,18,0],[18,18,0,0,1,0,1,0,0,0,0,18,0,0,18,18,1,0,0,0,1,0,0,18,0,0,0,0,0,18,0,0,0,0,1,18],[0,0,0,18,0,1,0,0,0,18,0,0,1,0,0,18,0,0,0,1,0,18,0,0,0,0,18,17,1,1,0,0,1,18,0,0],[12,17,17,5,2,2,2,14,17,5,0,0,0,0,17,5,2,14,0,0,17,5,0,2,0,0,3,12,14,14,0,0,10,3,2,2],[18,0,0,0,1,0,1,1,0,0,0,18,0,0,18,0,1,0,0,0,1,1,0,18,0,0,0,0,1,0,0,0,0,0,1,18] >;
 

He3.6D6 in GAP, Magma, Sage, TeX 

{\rm He}_3._6D_6
 
% in TeX
 
G:=Group("He3.6D6");
 
// GroupNames label
 
G:=SmallGroup(324,125);
 
// by ID
 
G=gap.SmallGroup(324,125);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,80,1593,453,2164,1096,3899]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^6=e*b*e=b^-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,d*c*d^-1=c^-1,c*e=e*c,e*d*e=b*d^5>;
 
// generators/relations
 

Export

Character table of He3.6D6 in TeX

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