Copied to
clipboard

G = C3×He3  order 81 = 34

Direct product of C3 and He3

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

Aliases: C3×He3, C33⋊2C3, C32⋊C32, C3.1C33, SmallGroup(81,12)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

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

Generators and relations for C3×He3
 G = < a,b,c,d | a3=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 104 in 56 conjugacy classes, 32 normal (4 characteristic)
C1, C3, C3, C3, C32, C32, C32, He3, C33, C3×He3
Quotients: C1, C3, C32, He3, C33, C3×He3

Permutation representations of C3×He3
►On 27 points - transitive group 27T18
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 9 16)(2 7 17)(3 8 18)(4 23 11)(5 24 12)(6 22 10)(13 27 20)(14 25 21)(15 26 19)

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

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

C3×He3 is a maximal subgroup of
 He3⋊4S3  He3⋊5S3  C32.24He3  C33.C32  C32.27He3  He3⋊C9  C32⋊He3  C9⋊He3  C32.23C33  C33⋊C32  He3.C32  He3⋊C32  3+ 1+4  3- 1+4
C3×He3 is a maximal quotient of
 C32⋊He3  C34.C3  C9⋊He3  C32.23C33  C9.He3  C33⋊C32  He3.C32  He3⋊C32  C32.C33  C9.2He3

33 conjugacy classes

class 1 3A···3H3I···3AF
order13···33···3
size11···13···3

33 irreducible representations

dim1113
type+
imageC1C3C3He3
kernelC3×He3He3C33C3
# reps11886

Matrix representation of C3×He3 ►in GL4(𝔽7) generated by

2000
0200
0020
0002
,
4000
0020
0552
0002
,
1000
0200
0020
0002
,
1000
0010
0334
0544
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,0,5,0,0,2,5,0,0,0,2,2],[1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,0,3,5,0,1,3,4,0,0,4,4] >;
 

C3×He3 in GAP, Magma, Sage, TeX 

C_3\times {\rm He}_3
 
% in TeX
 
G:=Group("C3xHe3");
 
// GroupNames label
 
G:=SmallGroup(81,12);
 
// by ID
 
G=gap.SmallGroup(81,12);
 
# by ID
 
G:=PCGroup([4,-3,3,3,-3,241]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁