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## G = C3×He3order 81 = 34

### Direct product of C3 and He3

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

Aliases: C3×He3, C332C3, 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
He34S3  He35S3  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 ··· 3H 3I ··· 3AF order 1 3 ··· 3 3 ··· 3 size 1 1 ··· 1 3 ··· 3

33 irreducible representations

 dim 1 1 1 3 type + image C1 C3 C3 He3 kernel C3×He3 He3 C33 C3 # reps 1 18 8 6

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

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

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