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## G = C9.4He3order 243 = 35

### 2nd central extension by C9 of He3

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

Aliases: C9.4He3, C33.2C9, C9.43- 1+2, C27⋊C31C3, (C3×C9).3C9, (C32×C9).9C3, C3.6(C32⋊C9), C32.10(C3×C9), (C3×C9).24C32, SmallGroup(243,16)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — C9.4He3
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C9.4He3
 Lower central C1 — C3 — C32 — C9.4He3
 Upper central C1 — C9 — C3×C9 — C9.4He3
 Jennings C1 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C3×C9 — C9.4He3

Generators and relations for C9.4He3
G = < a,b,c,d | a9=b3=c3=1, d3=a, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a3bc-1, dcd-1=a6c >

Permutation representations of C9.4He3
On 27 points - transitive group 27T92
Generators in S27
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)
(1 10 19)(2 20 11)(3 12 21)(4 13 22)(5 23 14)(6 15 24)(7 16 25)(8 26 17)(9 18 27)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 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)

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

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

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

G:=TransitiveGroup(27,92);

C9.4He3 is a maximal subgroup of   C33.D9

51 conjugacy classes

 class 1 3A 3B 3C ··· 3J 9A ··· 9F 9G ··· 9V 27A ··· 27R order 1 3 3 3 ··· 3 9 ··· 9 9 ··· 9 27 ··· 27 size 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3 9 ··· 9

51 irreducible representations

 dim 1 1 1 1 1 3 3 3 type + image C1 C3 C3 C9 C9 He3 3- 1+2 C9.4He3 kernel C9.4He3 C27⋊C3 C32×C9 C3×C9 C33 C9 C9 C1 # reps 1 6 2 12 6 2 4 18

Matrix representation of C9.4He3 in GL3(𝔽109) generated by

 66 0 0 0 66 0 0 0 66
,
 45 0 0 0 45 0 0 0 1
,
 1 0 0 0 63 0 0 0 45
,
 0 1 0 0 0 1 66 0 0
G:=sub<GL(3,GF(109))| [66,0,0,0,66,0,0,0,66],[45,0,0,0,45,0,0,0,1],[1,0,0,0,63,0,0,0,45],[0,0,66,1,0,0,0,1,0] >;

C9.4He3 in GAP, Magma, Sage, TeX

C_9._4{\rm He}_3
% in TeX

G:=Group("C9.4He3");
// GroupNames label

G:=SmallGroup(243,16);
// by ID

G=gap.SmallGroup(243,16);
# by ID

G:=PCGroup([5,-3,3,-3,3,-3,135,121,1352,78]);
// Polycyclic

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

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