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## G = C9.He3order 243 = 35

### 1st non-split extension by C9 of He3 acting via He3/C32=C3

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

Aliases: C9.1He3, He3.2C32, C32.5C33, C33.35C32, 3- 1+2.2C32, C3≀C33C3, C9○He31C3, (C32×C9)⋊8C3, He3.C33C3, C3.11(C3×He3), He3⋊C35C3, C3.He36C3, (C3×C9).29C32, SmallGroup(243,55)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C9.He3
G = < a,b,c,d | a9=b3=c3=d3=1, ab=ba, ac=ca, dad-1=a4, bc=cb, dbd-1=a3bc-1, dcd-1=a6c >

Subgroups: 153 in 67 conjugacy classes, 33 normal (11 characteristic)
C1, C3, C3, C9, C9, C9, C32, C32, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, C33, C3≀C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C9.He3
Quotients: C1, C3, C32, He3, C33, C3×He3, C9.He3

Permutation representations of C9.He3
On 27 points - transitive group 27T93
Generators in S27
(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)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)
(1 20 14)(2 27 18)(3 25 13)(4 23 17)(5 21 12)(6 19 16)(7 26 11)(8 24 15)(9 22 10)

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

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

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

G:=TransitiveGroup(27,93);

C9.He3 is a maximal subgroup of   C3≀S33C3  C3≀C3.S3  C3≀C3⋊S3

51 conjugacy classes

 class 1 3A 3B 3C ··· 3J 3K ··· 3P 9A ··· 9F 9G ··· 9V 9W ··· 9AH order 1 3 3 3 ··· 3 3 ··· 3 9 ··· 9 9 ··· 9 9 ··· 9 size 1 1 1 3 ··· 3 9 ··· 9 1 ··· 1 3 ··· 3 9 ··· 9

51 irreducible representations

 dim 1 1 1 1 1 1 1 3 3 type + image C1 C3 C3 C3 C3 C3 C3 He3 C9.He3 kernel C9.He3 C3≀C3 He3.C3 He3⋊C3 C3.He3 C32×C9 C9○He3 C9 C1 # reps 1 6 6 2 4 2 6 6 18

Matrix representation of C9.He3 in GL3(𝔽19) generated by

 9 0 0 2 4 0 15 0 6
,
 11 0 0 0 11 0 12 0 1
,
 1 0 0 9 7 0 7 0 11
,
 9 6 0 10 10 1 4 11 0
G:=sub<GL(3,GF(19))| [9,2,15,0,4,0,0,0,6],[11,0,12,0,11,0,0,0,1],[1,9,7,0,7,0,0,0,11],[9,10,4,6,10,11,0,1,0] >;

C9.He3 in GAP, Magma, Sage, TeX

C_9.{\rm He}_3
% in TeX

G:=Group("C9.He3");
// GroupNames label

G:=SmallGroup(243,55);
// by ID

G=gap.SmallGroup(243,55);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,301,147,2163]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^4,b*c=c*b,d*b*d^-1=a^3*b*c^-1,d*c*d^-1=a^6*c>;
// generators/relations

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