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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

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

Character table of C3.He3

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

 11 0 0 0 11 0 0 0 11
,
 4 0 0 0 4 0 16 0 6
,
 1 0 0 6 11 0 10 0 7
,
 6 10 0 12 13 1 12 15 0
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

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