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G = C3.He3order 81 = 34

4th central stem extension by C3 of He3

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

Aliases: C3.5He3, C32.4C32, 3- 1+2.C3, (C3×C9).2C3, SmallGroup(81,10)

Series: Derived Chief Lower central Upper central Jennings

C1C32 — C3.He3
C1C3C32C3×C9 — C3.He3
C1C3C32 — C3.He3
C1C3C32 — C3.He3
C1C3C32 — C3.He3

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

3C3
3C9
3C9
3C9
3C9

Character table of C3.He3

 class 13A3B3C3D9A9B9C9D9E9F9G9H9I9J9K9L
 size 11133333333999999
ρ111111111111111111    trivial
ρ211111ζ3ζ3ζ32ζ32ζ3ζ32ζ31ζ32ζ31ζ32    linear of order 3
ρ311111ζ3ζ3ζ32ζ32ζ3ζ32ζ32ζ32ζ31ζ31    linear of order 3
ρ411111ζ32ζ32ζ3ζ3ζ32ζ3ζ3ζ3ζ321ζ321    linear of order 3
ρ511111111111ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ611111111111ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ711111ζ32ζ32ζ3ζ3ζ32ζ31ζ321ζ3ζ3ζ32    linear of order 3
ρ811111ζ3ζ3ζ32ζ32ζ3ζ321ζ31ζ32ζ32ζ3    linear of order 3
ρ911111ζ32ζ32ζ3ζ3ζ32ζ3ζ321ζ3ζ321ζ3    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+3-3/2-3-3-3/200ζ97+2ζ9949ζ98+2ζ9598929794ζ95+2ζ92000000    complex faithful
ρ133-3-3-3/2-3+3-3/200ζ98+2ζ95ζ95+2ζ9297949499892ζ97+2ζ9000000    complex faithful
ρ143-3-3-3/2-3+3-3/200ζ95+2ζ929892ζ97+2ζ99794ζ98+2ζ95949000000    complex faithful
ρ153-3+3-3/2-3-3-3/2009499794ζ95+2ζ92ζ98+2ζ95ζ97+2ζ99892000000    complex faithful
ρ163-3+3-3/2-3-3-3/2009794ζ97+2ζ99892ζ95+2ζ92949ζ98+2ζ95000000    complex faithful
ρ173-3-3-3/2-3+3-3/2009892ζ98+2ζ95949ζ97+2ζ9ζ95+2ζ929794000000    complex faithful

Permutation representations of C3.He3
On 27 points - transitive group 27T25
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)
(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)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)
(1 26 11 7 23 17 4 20 14)(2 21 12 8 27 18 5 24 15)(3 25 13 9 22 10 6 19 16)

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

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

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

G:=TransitiveGroup(27,25);

C3.He3 is a maximal subgroup of
3- 1+2.S3  C92⋊C3  C92.C3  C32.He3  C32.5He3  C32.6He3  C9.He3  C32.C33  C9.2He3  C62.15C32  C62.C32
C3.He3 is a maximal quotient of
C33.3C32  C32.28He3  C32.29He3  C32.20He3  3- 1+2⋊C9  C62.15C32  C62.C32

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

1100
0110
0011
,
400
040
1606
,
100
6110
1007
,
6100
12131
12150
G:=sub<GL(3,GF(19))| [11,0,0,0,11,0,0,0,11],[4,0,16,0,4,0,0,0,6],[1,6,10,0,11,0,0,0,7],[6,12,12,10,13,15,0,1,0] >;

C3.He3 in GAP, Magma, Sage, TeX

C_3.{\rm He}_3
% in TeX

G:=Group("C3.He3");
// GroupNames label

G:=SmallGroup(81,10);
// by ID

G=gap.SmallGroup(81,10);
# by ID

G:=PCGroup([4,-3,3,-3,-3,108,97,149,434]);
// Polycyclic

G:=Group<a,b,c,d|a^3=c^3=1,b^3=a^-1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a*b*c^-1,d*c*d^-1=a^-1*c>;
// generators/relations

Export

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

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