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G = C32.He3  order 243 = 35

4th non-split extension by C32 of He3 acting via He3/C32=C3

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

Aliases: C32.4He3, C9⋊C9⋊1C3, (C3×C9).2C32, He3⋊C3⋊2C3, C3.He3⋊2C3, C3.9(He3⋊C3), SmallGroup(243,28)

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

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

Generators and relations for C32.He3
 G = < a,b,c,d,e | a3=b3=e3=1, c3=a-1, d3=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
27C3
C32
3C9
9C9
9C32
9C32
9C9
C3×C9
3He3
3He3
3C3×C9
33- 1+2
C9⋊C9
He3⋊C3
C3.He3
He3⋊C3
C32.He3

Character table of C32.He3

 class 13A3B3C3D3E3F3G3H9A9B9C9D9E9F9G9H9I9J
 size 1113327272727999999992727
ρ11111111111111111111    trivial
ρ2111111ζ3ζ32111ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ32    linear of order 3
ρ311111ζ3211ζ311ζ32ζ32ζ3ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ411111ζ32ζ32ζ3ζ311ζ3ζ3ζ32ζ32ζ3ζ3211    linear of order 3
ρ511111ζ311ζ3211ζ3ζ3ζ32ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ611111ζ3ζ3ζ32ζ3211ζ32ζ32ζ3ζ3ζ32ζ311    linear of order 3
ρ711111ζ32ζ3ζ32ζ311111111ζ32ζ3    linear of order 3
ρ811111ζ3ζ32ζ3ζ3211111111ζ3ζ32    linear of order 3
ρ9111111ζ32ζ3111ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ3    linear of order 3
ρ10333330000-3-3√-3/2-3+3√-3/200000000    complex lifted from He3
ρ11333330000-3+3√-3/2-3-3√-3/200000000    complex lifted from He3
ρ12333-3-3√-3/2-3+3√-3/2000000ζ98+2ζ952ζ98+ζ922ζ97+ζ94ζ97+2ζ9ζ95+2ζ922ζ94+ζ900    complex lifted from He3⋊C3
ρ13333-3-3√-3/2-3+3√-3/2000000ζ95+2ζ92ζ98+2ζ95ζ97+2ζ92ζ94+ζ92ζ98+ζ922ζ97+ζ9400    complex lifted from He3⋊C3
ρ14333-3+3√-3/2-3-3√-3/20000002ζ94+ζ9ζ97+2ζ9ζ95+2ζ922ζ98+ζ922ζ97+ζ94ζ98+2ζ9500    complex lifted from He3⋊C3
ρ15333-3-3√-3/2-3+3√-3/20000002ζ98+ζ92ζ95+2ζ922ζ94+ζ92ζ97+ζ94ζ98+2ζ95ζ97+2ζ900    complex lifted from He3⋊C3
ρ16333-3+3√-3/2-3-3√-3/20000002ζ97+ζ942ζ94+ζ92ζ98+ζ92ζ98+2ζ95ζ97+2ζ9ζ95+2ζ9200    complex lifted from He3⋊C3
ρ17333-3+3√-3/2-3-3√-3/2000000ζ97+2ζ92ζ97+ζ94ζ98+2ζ95ζ95+2ζ922ζ94+ζ92ζ98+ζ9200    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.He3
►On 27 points - transitive group 27T94
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 7 5 2 9 6 3 8 4)(10 11 15 13 14 18 16 17 12)(19 21 26 25 27 23 22 24 20)

(1 22 10)(2 19 13)(3 25 16)(4 26 12)(5 23 15)(6 20 18)(7 21 11)(8 24 17)(9 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), (1,7,5,2,9,6,3,8,4)(10,11,15,13,14,18,16,17,12)(19,21,26,25,27,23,22,24,20), (1,22,10)(2,19,13)(3,25,16)(4,26,12)(5,23,15)(6,20,18)(7,21,11)(8,24,17)(9,27,14)>;
 
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), (1,7,5,2,9,6,3,8,4)(10,11,15,13,14,18,16,17,12)(19,21,26,25,27,23,22,24,20), (1,22,10)(2,19,13)(3,25,16)(4,26,12)(5,23,15)(6,20,18)(7,21,11)(8,24,17)(9,27,14) );
 
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)], [(1,7,5,2,9,6,3,8,4),(10,11,15,13,14,18,16,17,12),(19,21,26,25,27,23,22,24,20)], [(1,22,10),(2,19,13),(3,25,16),(4,26,12),(5,23,15),(6,20,18),(7,21,11),(8,24,17),(9,27,14)]])
 
G:=TransitiveGroup(27,94);
 

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

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

100000000
010000000
001000000
000700000
000070000
000007000
0000001100
0000000110
0000000011
,
700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
100000000
18110000000
1107000000
000190000
00018187000
0000120000
000000104
000000708
0000000718
,
760000000
0121000000
880000000
000760000
0000121000
000880000
0000001140
000000087
00000018180
,
000100000
000010000
000001000
000000100
000000010
000000001
100000000
010000000
001000000

G:=sub<GL(9,GF(19))| [1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[1,18,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,9,18,12,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,4,8,18],[7,0,8,0,0,0,0,0,0,6,12,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7,0,8,0,0,0,0,0,0,6,12,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11,0,18,0,0,0,0,0,0,4,8,18,0,0,0,0,0,0,0,7,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C32.He3 in GAP, Magma, Sage, TeX 

C_3^2.{\rm He}_3
 
% in TeX
 
G:=Group("C3^2.He3");
 
// GroupNames label
 
G:=SmallGroup(243,28);
 
// by ID
 
G=gap.SmallGroup(243,28);
 
# by ID
 
G:=PCGroup([5,-3,3,-3,-3,-3,121,186,542,457,282,2163]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=e^3=1,c^3=a^-1,d^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.He3 in TeX
Character table of C32.He3 in TeX

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