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## G = C32.5He3order 243 = 35

### 5th non-split extension by C32 of He3 acting via He3/C32=C3

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

Aliases: C32.5He3, C9⋊C9.2C3, (C3×C9).3C32, C3.He3.2C3, C3.10(He3⋊C3), SmallGroup(243,29)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3×C9 — C32.5He3
 Chief series C1 — C3 — C32 — C3×C9 — C9⋊C9 — C32.5He3
 Lower central C1 — C3 — C32 — C3×C9 — C32.5He3
 Upper central C1 — C3 — C32 — C3×C9 — C32.5He3
 Jennings C1 — C3 — C3 — C3 — C32 — C3×C9 — C32.5He3

Generators and relations for C32.5He3
G = < a,b,c,d,e | a3=b3=1, c3=a-1, d3=b-1, e3=b, 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 >

Character table of C32.5He3

 class 1 3A 3B 3C 3D 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 9M 9N size 1 1 1 3 3 9 9 9 9 9 9 9 9 27 27 27 27 27 27 ρ1 1 1 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 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ3 1 1 1 1 1 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 linear of order 3 ρ4 1 1 1 1 1 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 linear of order 3 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ7 1 1 1 1 1 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ32 1 linear of order 3 ρ8 1 1 1 1 1 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ3 1 linear of order 3 ρ9 1 1 1 1 1 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ10 3 3 3 3 3 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ11 3 3 3 3 3 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ12 3 3 3 -3+3√-3/2 -3-3√-3/2 ζ95+2ζ92 0 0 2ζ97+ζ94 2ζ94+ζ9 2ζ98+ζ92 ζ98+2ζ95 ζ97+2ζ9 0 0 0 0 0 0 complex lifted from He3⋊C3 ρ13 3 3 3 -3+3√-3/2 -3-3√-3/2 2ζ98+ζ92 0 0 ζ97+2ζ9 2ζ97+ζ94 ζ98+2ζ95 ζ95+2ζ92 2ζ94+ζ9 0 0 0 0 0 0 complex lifted from He3⋊C3 ρ14 3 3 3 -3+3√-3/2 -3-3√-3/2 ζ98+2ζ95 0 0 2ζ94+ζ9 ζ97+2ζ9 ζ95+2ζ92 2ζ98+ζ92 2ζ97+ζ94 0 0 0 0 0 0 complex lifted from He3⋊C3 ρ15 3 3 3 -3-3√-3/2 -3+3√-3/2 2ζ97+ζ94 0 0 ζ95+2ζ92 ζ98+2ζ95 ζ97+2ζ9 2ζ94+ζ9 2ζ98+ζ92 0 0 0 0 0 0 complex lifted from He3⋊C3 ρ16 3 3 3 -3-3√-3/2 -3+3√-3/2 2ζ94+ζ9 0 0 ζ98+2ζ95 2ζ98+ζ92 2ζ97+ζ94 ζ97+2ζ9 ζ95+2ζ92 0 0 0 0 0 0 complex lifted from He3⋊C3 ρ17 3 3 3 -3-3√-3/2 -3+3√-3/2 ζ97+2ζ9 0 0 2ζ98+ζ92 ζ95+2ζ92 2ζ94+ζ9 2ζ97+ζ94 ζ98+2ζ95 0 0 0 0 0 0 complex lifted from He3⋊C3 ρ18 9 -9+9√-3/2 -9-9√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ19 9 -9-9√-3/2 -9+9√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C32.5He3
On 27 points - transitive group 27T90
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 25 13 3 19 10 2 22 16)(4 20 15 6 23 12 5 26 18)(7 24 14 8 27 11 9 21 17)

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

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

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

G:=TransitiveGroup(27,90);

C32.5He3 is a maximal subgroup of   C9⋊C9.3S3

Matrix representation of C32.5He3 in GL9(𝔽19)

 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 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 8 11 7 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 18 8 4 0 0 0 0 0 0 7 1 11 0 0 0 0 0 0 0 0 0 18 8 4 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 1
,
 0 1 0 0 0 0 0 0 0 8 12 6 0 0 0 0 0 0 8 0 7 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 8 12 6 0 0 0 0 0 0 8 0 7 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 18 8 4 0 0 0 0 0 0 18 0 11
,
 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 0 0 0 0 0 0 1 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

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],[1,0,8,0,0,0,0,0,0,0,11,11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,18,7,0,0,0,0,0,0,11,8,1,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,0,18,11,11,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,4,0,1],[0,8,8,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,7,8,0,0,0,0,0,0,0,0,4,11],[0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0] >;

C32.5He3 in GAP, Magma, Sage, TeX

C_3^2._5{\rm He}_3
% in TeX

G:=Group("C3^2.5He3");
// GroupNames label

G:=SmallGroup(243,29);
// by ID

G=gap.SmallGroup(243,29);
# by ID

G:=PCGroup([5,-3,3,-3,-3,-3,405,121,186,542,457,282,2163]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=1,c^3=a^-1,d^3=b^-1,e^3=b,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

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