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## G = He3.C32order 243 = 35

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

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

Aliases: He3.4C32, C32.14He3, C32.7C33, C33.13C32, 3- 1+24C32, (C3×C9)⋊2C32, He3.C31C3, (C3×He3).7C3, C3.13(C3×He3), (C3×3- 1+2)⋊8C3, SmallGroup(243,57)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — He3.C32
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — He3.C32
 Lower central C1 — C3 — C32 — He3.C32
 Upper central C1 — C3 — C33 — He3.C32
 Jennings C1 — C3 — C32 — He3.C32

Generators and relations for He3.C32
G = < a,b,c,d,e | a3=b3=c3=e3=1, d3=b, ab=ba, cac-1=ab-1, ad=da, ae=ea, bc=cb, ede-1=bd=db, be=eb, dcd-1=ab-1c, ce=ec >

Subgroups: 180 in 68 conjugacy classes, 33 normal (8 characteristic)
C1, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, He3.C3, C3×He3, C3×3- 1+2, C3×3- 1+2, He3.C32
Quotients: C1, C3, C32, He3, C33, C3×He3, He3.C32

Permutation representations of He3.C32
On 27 points - transitive group 27T99
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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)

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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)>;

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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24) );

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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,22,25),(21,27,24)]])

G:=TransitiveGroup(27,99);

On 27 points - transitive group 27T109
Generators in S27
(1 20 15)(2 21 16)(3 22 17)(4 23 18)(5 24 10)(6 25 11)(7 26 12)(8 27 13)(9 19 14)
(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 10)(3 17 19)(5 24 13)(6 11 22)(8 27 16)(9 14 25)(12 18 15)(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)
(1 4 7)(3 9 6)(11 17 14)(12 15 18)(19 25 22)(20 23 26)

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

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

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

G:=TransitiveGroup(27,109);

He3.C32 is a maximal subgroup of   He3.C3⋊C6  C9⋊S3⋊C32  He3.C32C6

35 conjugacy classes

 class 1 3A 3B 3C ··· 3J 3K ··· 3P 9A ··· 9R order 1 3 3 3 ··· 3 3 ··· 3 9 ··· 9 size 1 1 1 3 ··· 3 9 ··· 9 9 ··· 9

35 irreducible representations

 dim 1 1 1 1 3 9 type + image C1 C3 C3 C3 He3 He3.C32 kernel He3.C32 He3.C3 C3×He3 C3×3- 1+2 C32 C1 # reps 1 18 2 6 6 2

Matrix representation of He3.C32 in GL9(𝔽19)

 0 1 0 0 0 0 0 0 0 12 8 6 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 8 6 0 0 0 0 0 0 0 0 11 0 0 0 11 12 0 1 8 0 7 6 0 0 0 7 0 0 11 0 12 1 0 0 18 0 0 12 0 8 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 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 12 1 7 0 0 0 0 0 0 0 0 0 8 18 4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 7 11 0 0 0 7 18 0 11 12 0 1 9 0 4 12 0 16 7 12 18 18 7 1 0 0 6 11 1 11 12 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 11 12 0 1 8 0 7 6 0 0 0 0 0 0 0 0 11 1 7 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 11 0 1 8 7 0 0 0 0 12 0
,
 11 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 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 18 11 0 1 8 0 0 7 0 11 12 0 8 7 0 0 0 7

G:=sub<GL(9,GF(19))| [0,12,0,0,0,0,11,0,0,1,8,0,0,0,0,12,0,0,0,6,11,0,0,0,0,7,18,0,0,0,0,12,0,1,0,0,0,0,0,1,8,0,8,0,0,0,0,0,0,6,11,0,11,12,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,6,12,8,0,0,0,0,0,0,0,1,0],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[1,0,12,0,0,0,7,4,1,0,11,1,0,0,0,18,12,0,0,0,7,0,0,0,0,0,0,0,0,0,8,1,8,11,16,6,0,0,0,18,0,7,12,7,11,0,0,0,4,0,11,0,12,1,0,0,0,0,0,0,1,18,11,0,0,0,0,0,0,9,18,12,0,0,0,0,0,0,0,7,0],[0,0,0,0,11,0,7,12,1,0,0,0,0,12,0,0,1,8,0,0,0,0,0,0,0,0,7,1,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,6,11,0,11,12,0,0,0,0,0,1,0,0,0],[11,0,0,0,0,0,0,18,11,0,11,0,0,0,0,0,11,12,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,1,8,0,0,0,0,1,0,0,8,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7] >;

He3.C32 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(243,57);
// by ID

G=gap.SmallGroup(243,57);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,301,276,457,2163]);
// Polycyclic

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

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