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G = He3.C3  order 81 = 34

The non-split extension by He3 of C3 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: He3.C3, C3.3He3, C32.2C32, 3- 1+2⋊2C3, (C3×C9)⋊2C3, SmallGroup(81,8)

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

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

Generators and relations for He3.C3
 G = < a,b,c,d | a3=b3=c3=1, d3=b, ab=ba, cac-1=ab-1, ad=da, bc=cb, bd=db, dcd-1=ab-1c >

C1
C3
3C3
9C3
C32
3C9
3C32
3C9
3C9
C3×C9
3- 1+2
3- 1+2
He3
He3.C3

Character table of He3.C3

 class 13A3B3C3D3E3F9A9B9C9D9E9F9G9H9I9J
 size 11133993333339999
ρ111111111111111111    trivial
ρ211111ζ3ζ32111111ζ3ζ3ζ32ζ32    linear of order 3
ρ311111ζ32ζ3111111ζ32ζ32ζ3ζ3    linear of order 3
ρ411111ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ31ζ321ζ3    linear of order 3
ρ51111111ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ611111ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ321ζ31    linear of order 3
ρ711111ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ321ζ31ζ32    linear of order 3
ρ811111ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ31ζ321    linear of order 3
ρ91111111ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ10333-3-3√-3/2-3+3√-3/2000000000000    complex lifted from He3
ρ11333-3+3√-3/2-3-3√-3/2000000000000    complex lifted from He3
ρ123-3+3√-3/2-3-3√-3/20000ζ97+2ζ94ζ94+2ζ9ζ98+2ζ922ζ98+ζ952ζ97+ζ92ζ95+ζ920000    complex faithful
ρ133-3+3√-3/2-3-3√-3/20000ζ94+2ζ92ζ97+ζ92ζ95+ζ92ζ98+2ζ92ζ97+2ζ942ζ98+ζ950000    complex faithful
ρ143-3+3√-3/2-3-3√-3/200002ζ97+ζ9ζ97+2ζ942ζ98+ζ952ζ95+ζ92ζ94+2ζ9ζ98+2ζ920000    complex faithful
ρ153-3-3√-3/2-3+3√-3/200002ζ95+ζ922ζ98+ζ952ζ97+ζ9ζ94+2ζ9ζ98+2ζ92ζ97+2ζ940000    complex faithful
ρ163-3-3√-3/2-3+3√-3/20000ζ98+2ζ922ζ95+ζ92ζ94+2ζ9ζ97+2ζ942ζ98+ζ952ζ97+ζ90000    complex faithful
ρ173-3-3√-3/2-3+3√-3/200002ζ98+ζ95ζ98+2ζ92ζ97+2ζ942ζ97+ζ92ζ95+ζ92ζ94+2ζ90000    complex faithful

Permutation representations of He3.C3
►On 27 points - transitive group 27T20
Generators in S27
(1 20 13)(2 21 14)(3 22 15)(4 23 16)(5 24 17)(6 25 18)(7 26 10)(8 27 11)(9 19 12)

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

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

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

►On 27 points - transitive group 27T26
Generators in S27
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)

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

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

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

He3.C3 is a maximal subgroup of
 He3.C6  He3.S3  He3.3S3  C9.He3  He3.C32  C9.2He3  C62.13C32  He3.A4  3- 1+2⋊A4
He3.C3 is a maximal quotient of
 C33.C32  C33.3C32  C32.27He3  C32.29He3  C33.7C32  C32.19He3  He3⋊C9  3- 1+2⋊C9  C62.13C32  He3.A4  3- 1+2⋊A4

Matrix representation of He3.C3 ►in GL3(𝔽19) generated by

1160
081
0120
,
700
070
007
,
100
12110
707
,
91315
040
004
G:=sub<GL(3,GF(19))| [11,0,0,6,8,12,0,1,0],[7,0,0,0,7,0,0,0,7],[1,12,7,0,11,0,0,0,7],[9,0,0,13,4,0,15,0,4] >;
 

He3.C3 in GAP, Magma, Sage, TeX 

{\rm He}_3.C_3
 
% in TeX
 
G:=Group("He3.C3");
 
// GroupNames label
 
G:=SmallGroup(81,8);
 
// by ID
 
G=gap.SmallGroup(81,8);
 
# by ID
 
G:=PCGroup([4,-3,3,-3,-3,97,77,434]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^3=b,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*b^-1*c>;
 
// generators/relations
 

Export

Subgroup lattice of He3.C3 in TeX
Character table of He3.C3 in TeX

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