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## G = C9○He3order 81 = 34

### Central product of C9 and He3

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C9He3, C9.C32, He3.2C3, C3.3C33, C93- 1+2, C32.5C32, 3- 1+23C3, (C3×C9)⋊5C3, SmallGroup(81,14)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C9○He3
G = < a,b,c,d | a9=b3=d3=1, c1=a6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a3b, cd=dc >

Permutation representations of C9○He3
On 27 points - transitive group 27T28
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 26 11)(2 27 12)(3 19 13)(4 20 14)(5 21 15)(6 22 16)(7 23 17)(8 24 18)(9 25 10)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)
(1 17 23)(2 18 24)(3 10 25)(4 11 26)(5 12 27)(6 13 19)(7 14 20)(8 15 21)(9 16 22)

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

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

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

G:=TransitiveGroup(27,28);

C9○He3 is a maximal subgroup of
He3.4S3  He3.4C6  C9.5He3  C9.6He3  C27○He3  C9.He3  C9.2He3  3- 1+4  C62.25C32  He3.2A4  C62.9C32
C9○He3 is a maximal quotient of
C923C3  C9×He3  C9×3- 1+2  C9⋊He3  C32.23C33  C9⋊3- 1+2  C33.31C32  C927C3  C924C3  C925C3  C928C3  C62.25C32  He3.2A4  C62.9C32

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 3 type + image C1 C3 C3 C3 C9○He3 kernel C9○He3 C3×C9 He3 3- 1+2 C1 # reps 1 8 2 16 6

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

 17 0 0 0 17 0 0 0 17
,
 1 0 0 0 7 0 0 0 11
,
 7 0 0 0 7 0 0 0 7
,
 0 7 0 0 0 11 1 0 0
G:=sub<GL(3,GF(19))| [17,0,0,0,17,0,0,0,17],[1,0,0,0,7,0,0,0,11],[7,0,0,0,7,0,0,0,7],[0,0,1,7,0,0,0,11,0] >;

C9○He3 in GAP, Magma, Sage, TeX

C_9\circ {\rm He}_3
% in TeX

G:=Group("C9oHe3");
// GroupNames label

G:=SmallGroup(81,14);
// by ID

G=gap.SmallGroup(81,14);
# by ID

G:=PCGroup([4,-3,3,3,-3,241,46]);
// Polycyclic

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

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