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## G = He3.C3order 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+22C3, (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 >

Character table of He3.C3

 class 1 3A 3B 3C 3D 3E 3F 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J size 1 1 1 3 3 9 9 3 3 3 3 3 3 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ3 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ4 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 1 ζ32 1 ζ3 linear of order 3 ρ5 1 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 1 ζ3 1 linear of order 3 ρ7 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 1 ζ3 1 ζ32 linear of order 3 ρ8 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 1 ζ32 1 linear of order 3 ρ9 1 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ10 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ11 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ12 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ97+2ζ94 ζ94+2ζ9 ζ98+2ζ92 2ζ98+ζ95 2ζ97+ζ9 2ζ95+ζ92 0 0 0 0 complex faithful ρ13 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ94+2ζ9 2ζ97+ζ9 2ζ95+ζ92 ζ98+2ζ92 ζ97+2ζ94 2ζ98+ζ95 0 0 0 0 complex faithful ρ14 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 2ζ97+ζ9 ζ97+2ζ94 2ζ98+ζ95 2ζ95+ζ92 ζ94+2ζ9 ζ98+2ζ92 0 0 0 0 complex faithful ρ15 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 2ζ95+ζ92 2ζ98+ζ95 2ζ97+ζ9 ζ94+2ζ9 ζ98+2ζ92 ζ97+2ζ94 0 0 0 0 complex faithful ρ16 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ98+2ζ92 2ζ95+ζ92 ζ94+2ζ9 ζ97+2ζ94 2ζ98+ζ95 2ζ97+ζ9 0 0 0 0 complex faithful ρ17 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 2ζ98+ζ95 ζ98+2ζ92 ζ97+2ζ94 2ζ97+ζ9 2ζ95+ζ92 ζ94+2ζ9 0 0 0 0 complex faithful

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

 11 6 0 0 8 1 0 12 0
,
 7 0 0 0 7 0 0 0 7
,
 1 0 0 12 11 0 7 0 7
,
 9 13 15 0 4 0 0 0 4
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

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