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G = C32.6He3  order 243 = 35

6th non-split extension by C32 of He3 acting via He3/C32=C3

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C32.6He3, C9⋊C9⋊2C3, (C3×C9).4C32, C3.He3⋊3C3, He3⋊C3.2C3, C3.11(He3⋊C3), 3-Sylow(2A(2,8).C3), SmallGroup(243,30)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C3×C9 — C32.6He3
Chief series
C1 — C3 — C32 — C3×C9 — C9⋊C9 — C32.6He3
Lower central
C1 — C3 — C32 — C3×C9 — C32.6He3
Upper central
C1 — C3 — C32 — C3×C9 — C32.6He3
Jennings
C1 — C3 — C3 — C3 — C32 — C3×C9 — C32.6He3

Generators and relations for C32.6He3
 G = < a,b,c,d,e | a3=b3=1, c3=a-1, d3=e3=b-1, ab=ba, ac=ca, ad=da, eae-1=ab-1, dcd-1=bc=cb, bd=db, be=eb, ece-1=acd2, ede-1=a-1bd >

C1
C3
3C3
27C3
C32
3C9
9C32
9C9
9C9
9C9
C3×C9
33- 1+2
33- 1+2
3C3×C9
3He3
C9⋊C9
C3.He3
He3⋊C3
C3.He3
C32.6He3

Character table of C32.6He3

 class 13A3B3C3D3E3F9A9B9C9D9E9F9G9H9I9J9K9L
 size 1113327279999999927272727
ρ11111111111111111111    trivial
ρ21111111ζ311ζ32ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ311111ζ3ζ32ζ311ζ32ζ32ζ3ζ3ζ321ζ32ζ31    linear of order 3
ρ41111111ζ3211ζ3ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ511111ζ3ζ3211111111ζ3ζ3ζ32ζ32    linear of order 3
ρ611111ζ32ζ3ζ311ζ32ζ32ζ3ζ3ζ32ζ311ζ32    linear of order 3
ρ711111ζ32ζ311111111ζ32ζ32ζ3ζ3    linear of order 3
ρ811111ζ3ζ32ζ3211ζ3ζ3ζ32ζ32ζ3ζ3211ζ3    linear of order 3
ρ911111ζ32ζ3ζ3211ζ3ζ3ζ32ζ32ζ31ζ3ζ321    linear of order 3
ρ1033333000-3+3√-3/2-3-3√-3/2000000000    complex lifted from He3
ρ1133333000-3-3√-3/2-3+3√-3/2000000000    complex lifted from He3
ρ12333-3-3√-3/2-3+3√-3/200ζ97+2ζ9002ζ98+ζ92ζ95+2ζ922ζ94+ζ92ζ97+ζ94ζ98+2ζ950000    complex lifted from He3⋊C3
ρ13333-3-3√-3/2-3+3√-3/2002ζ97+ζ9400ζ95+2ζ92ζ98+2ζ95ζ97+2ζ92ζ94+ζ92ζ98+ζ920000    complex lifted from He3⋊C3
ρ14333-3-3√-3/2-3+3√-3/2002ζ94+ζ900ζ98+2ζ952ζ98+ζ922ζ97+ζ94ζ97+2ζ9ζ95+2ζ920000    complex lifted from He3⋊C3
ρ15333-3+3√-3/2-3-3√-3/200ζ95+2ζ92002ζ97+ζ942ζ94+ζ92ζ98+ζ92ζ98+2ζ95ζ97+2ζ90000    complex lifted from He3⋊C3
ρ16333-3+3√-3/2-3-3√-3/2002ζ98+ζ9200ζ97+2ζ92ζ97+ζ94ζ98+2ζ95ζ95+2ζ922ζ94+ζ90000    complex lifted from He3⋊C3
ρ17333-3+3√-3/2-3-3√-3/200ζ98+2ζ95002ζ94+ζ9ζ97+2ζ9ζ95+2ζ922ζ98+ζ922ζ97+ζ940000    complex lifted from He3⋊C3
ρ189-9+9√-3/2-9-9√-3/20000000000000000    complex faithful
ρ199-9-9√-3/2-9+9√-3/20000000000000000    complex faithful

Permutation representations of C32.6He3
►On 27 points - transitive group 27T91
Generators in S27
(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)

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

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

(1 19 13 2 25 16 3 22 10)(4 23 15 5 20 18 6 26 12)(7 24 17 9 21 11 8 27 14)
 
G:=sub<Sym(27)| (10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (1,3,2)(4,6,5)(7,8,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (4,5,6)(7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,8,5,2,7,6,3,9,4)(10,11,15,13,14,18,16,17,12)(19,21,26,25,27,23,22,24,20), (1,19,13,2,25,16,3,22,10)(4,23,15,5,20,18,6,26,12)(7,24,17,9,21,11,8,27,14)>;
 
G:=Group( (10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (1,3,2)(4,6,5)(7,8,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (4,5,6)(7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,8,5,2,7,6,3,9,4)(10,11,15,13,14,18,16,17,12)(19,21,26,25,27,23,22,24,20), (1,19,13,2,25,16,3,22,10)(4,23,15,5,20,18,6,26,12)(7,24,17,9,21,11,8,27,14) );
 
G=PermutationGroup([[(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24)], [(1,3,2),(4,6,5),(7,8,9),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27)], [(4,5,6),(7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,8,5,2,7,6,3,9,4),(10,11,15,13,14,18,16,17,12),(19,21,26,25,27,23,22,24,20)], [(1,19,13,2,25,16,3,22,10),(4,23,15,5,20,18,6,26,12),(7,24,17,9,21,11,8,27,14)]])
 
G:=TransitiveGroup(27,91);
 

C32.6He3 is a maximal subgroup of   C9⋊C9⋊S3

Matrix representation of C32.6He3 ►in GL9(𝔽19)

100000000
010000000
001000000
0001100000
0000110000
0000011000
000000700
1870780070
780810007
,
1100000000
0110000000
0011000000
0001100000
0000110000
0000011000
0000001100
0000000110
0000000011
,
100000000
070000000
8111000000
000070000
000121815000
0001171000
8108101015
111100121111012
181800111801118
,
010000000
81210000000
1207000000
000010000
00081210000
0001207000
7807807150
111811118101211
187121871218180
,
000100000
000010000
000001000
000000100
111801118011100
000000071
700000000
1120000010
121170000120

G:=sub<GL(9,GF(19))| [1,0,0,0,0,0,0,18,7,0,1,0,0,0,0,0,7,8,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,7,8,0,0,0,0,11,0,0,8,1,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[1,0,8,0,0,0,8,11,18,0,7,1,0,0,0,1,11,18,0,0,11,0,0,0,0,0,0,0,0,0,0,12,11,8,0,0,0,0,0,7,18,7,1,12,11,0,0,0,0,15,1,0,11,18,0,0,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,15,12,18],[0,8,12,0,0,0,7,11,18,1,12,0,0,0,0,8,18,7,0,10,7,0,0,0,0,1,12,0,0,0,0,8,12,7,11,18,0,0,0,1,12,0,8,18,7,0,0,0,0,10,7,0,1,12,0,0,0,0,0,0,7,0,18,0,0,0,0,0,0,15,12,18,0,0,0,0,0,0,0,11,0],[0,0,0,0,11,0,7,1,12,0,0,0,0,18,0,0,12,11,0,0,0,0,0,0,0,0,7,1,0,0,0,11,0,0,0,0,0,1,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,11,0,0,0,0,0,0,0,0,10,7,0,1,12,0,0,0,0,0,1,0,0,0] >;
 

C32.6He3 in GAP, Magma, Sage, TeX 

C_3^2._6{\rm He}_3
 
% in TeX
 
G:=Group("C3^2.6He3");
 
// GroupNames label
 
G:=SmallGroup(243,30);
 
// by ID
 
G=gap.SmallGroup(243,30);
 
# by ID
 
G:=PCGroup([5,-3,3,-3,-3,-3,810,121,186,542,457,282,2163]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=1,c^3=a^-1,d^3=e^3=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,d*c*d^-1=b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=a*c*d^2,e*d*e^-1=a^-1*b*d>;
 
// generators/relations
 

Export

Subgroup lattice of C32.6He3 in TeX
Character table of C32.6He3 in TeX

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