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## G = C32.He3order 243 = 35

### 4th non-split extension by C32 of He3 acting via He3/C32=C3

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

Aliases: C32.4He3, C9⋊C91C3, (C3×C9).2C32, He3⋊C32C3, C3.He32C3, C3.9(He3⋊C3), SmallGroup(243,28)

Series: Derived Chief Lower central Upper central Jennings

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

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

 class 1 3A 3B 3C 3D 3E 3F 3G 3H 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J size 1 1 1 3 3 27 27 27 27 9 9 9 9 9 9 9 9 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 1 ζ3 ζ32 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ3 1 1 1 1 1 ζ32 1 1 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 1 1 linear of order 3 ρ5 1 1 1 1 1 ζ3 1 1 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 1 1 linear of order 3 ρ7 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ32 ζ3 linear of order 3 ρ8 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ3 ζ32 linear of order 3 ρ9 1 1 1 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ10 3 3 3 3 3 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 complex lifted from He3 ρ11 3 3 3 3 3 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 complex lifted from He3 ρ12 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ98+2ζ95 2ζ98+ζ92 2ζ97+ζ94 ζ97+2ζ9 ζ95+2ζ92 2ζ94+ζ9 0 0 complex lifted from He3⋊C3 ρ13 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ95+2ζ92 ζ98+2ζ95 ζ97+2ζ9 2ζ94+ζ9 2ζ98+ζ92 2ζ97+ζ94 0 0 complex lifted from He3⋊C3 ρ14 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 2ζ94+ζ9 ζ97+2ζ9 ζ95+2ζ92 2ζ98+ζ92 2ζ97+ζ94 ζ98+2ζ95 0 0 complex lifted from He3⋊C3 ρ15 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 2ζ98+ζ92 ζ95+2ζ92 2ζ94+ζ9 2ζ97+ζ94 ζ98+2ζ95 ζ97+2ζ9 0 0 complex lifted from He3⋊C3 ρ16 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 2ζ97+ζ94 2ζ94+ζ9 2ζ98+ζ92 ζ98+2ζ95 ζ97+2ζ9 ζ95+2ζ92 0 0 complex lifted from He3⋊C3 ρ17 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ97+2ζ9 2ζ97+ζ94 ζ98+2ζ95 ζ95+2ζ92 2ζ94+ζ9 2ζ98+ζ92 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.He3
On 27 points - transitive group 27T94
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 24 23 25 21 20 22 27 26)
(1 26 10)(2 23 13)(3 20 16)(4 21 12)(5 27 15)(6 24 18)(7 25 11)(8 19 17)(9 22 14)

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

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

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

G:=TransitiveGroup(27,94);

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

Matrix representation of C32.He3 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 18 11 0 0 0 0 0 0 0 11 0 7 0 0 0 0 0 0 0 0 0 1 9 0 0 0 0 0 0 0 18 18 7 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 4 0 0 0 0 0 0 7 0 8 0 0 0 0 0 0 0 7 18
,
 7 6 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 8 8 0 0 0 0 0 0 0 0 0 0 7 6 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 8 8 0 0 0 0 0 0 0 0 0 0 11 4 0 0 0 0 0 0 0 0 8 7 0 0 0 0 0 0 18 18 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 0 0 0 0 0 0 1 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

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],[1,18,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,9,18,12,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,4,8,18],[7,0,8,0,0,0,0,0,0,6,12,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7,0,8,0,0,0,0,0,0,6,12,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11,0,18,0,0,0,0,0,0,4,8,18,0,0,0,0,0,0,0,7,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0] >;

C32.He3 in GAP, Magma, Sage, TeX

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

G:=Group("C3^2.He3");
// GroupNames label

G:=SmallGroup(243,28);
// by ID

G=gap.SmallGroup(243,28);
# by ID

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

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