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## G = C32⋊He3order 243 = 35

### The semidirect product of C32 and He3 acting via He3/C32=C3

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

Aliases: C32⋊He3, C341C3, C331C32, C32.20C33, (C3×He3)⋊2C3, C3.3(C3×He3), SmallGroup(243,37)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — C32⋊He3
 Chief series C1 — C3 — C32 — C33 — C34 — C32⋊He3
 Lower central C1 — C32 — C32⋊He3
 Upper central C1 — C32 — C32⋊He3
 Jennings C1 — C32 — C32⋊He3

Generators and relations for C32⋊He3
G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 531 in 189 conjugacy classes, 45 normal (4 characteristic)
C1, C3, C3, C32, C32, C32, He3, C33, C33, C3×He3, C34, C32⋊He3
Quotients: C1, C3, C32, He3, C33, C3×He3, C32⋊He3

Permutation representations of C32⋊He3
On 27 points - transitive group 27T95
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 15 11)(2 13 12)(3 14 10)(4 26 8)(5 27 9)(6 25 7)(16 24 20)(17 22 21)(18 23 19)
(1 3 2)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 24 20)(17 22 21)(18 23 19)(25 26 27)
(1 13 10)(2 14 11)(3 15 12)(4 9 25)(5 7 26)(6 8 27)(16 17 18)(19 20 21)(22 23 24)
(1 8 17)(2 5 19)(3 25 24)(4 22 15)(6 16 10)(7 20 14)(9 23 12)(11 26 21)(13 27 18)

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

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

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

G:=TransitiveGroup(27,95);

C32⋊He3 is a maximal subgroup of   C34⋊C6  C34⋊S3  C343S3  C344C6  C346S3

51 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3AF 3AG ··· 3AX order 1 3 ··· 3 3 ··· 3 3 ··· 3 size 1 1 ··· 1 3 ··· 3 9 ··· 9

51 irreducible representations

 dim 1 1 1 3 type + image C1 C3 C3 He3 kernel C32⋊He3 C3×He3 C34 C32 # reps 1 24 2 24

Matrix representation of C32⋊He3 in GL6(𝔽7)

 1 0 0 0 0 0 0 2 0 0 0 0 1 6 4 0 0 0 0 0 0 1 1 3 0 0 0 0 2 0 0 0 0 0 0 4
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 1 0 0 0 0 0 0 2 0 0 0 0 1 6 4 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 5 3 1 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 6 6

G:=sub<GL(6,GF(7))| [1,0,1,0,0,0,0,2,6,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,1,2,0,0,0,0,3,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,1,0,0,0,0,2,6,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,1,3,0,0,0,0,0,1,4,0,0,0,0,0,0,1,0,1,0,0,0,0,0,6,0,0,0,0,1,6] >;

C32⋊He3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(243,37);
// by ID

G=gap.SmallGroup(243,37);
# by ID

G:=PCGroup([5,-3,3,3,-3,3,301,1352]);
// Polycyclic

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

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