Copied to
clipboard

## G = He3⋊C2order 54 = 2·33

### 2nd semidirect product of He3 and C2 acting faithfully

Aliases: He32C2, C322S3, C3.2(C3⋊S3), Aut(3- 1+2), SmallGroup(54,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C2
 Chief series C1 — C3 — C32 — He3 — He3⋊C2
 Lower central He3 — He3⋊C2
 Upper central C1 — C3

Generators and relations for He3⋊C2
G = < a,b,c,d | a3=b3=c3=d2=1, ab=ba, cac-1=ab-1, dad=a-1, bc=cb, bd=db, dcd=c-1 >

Character table of He3⋊C2

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B size 1 9 1 1 6 6 6 6 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ3 2 0 2 2 -1 -1 2 -1 0 0 orthogonal lifted from S3 ρ4 2 0 2 2 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ5 2 0 2 2 -1 -1 -1 2 0 0 orthogonal lifted from S3 ρ6 2 0 2 2 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ7 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ3 ζ32 complex faithful ρ8 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ32 ζ3 complex faithful ρ9 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ6 ζ65 complex faithful ρ10 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ65 ζ6 complex faithful

Permutation representations of He3⋊C2
On 9 points - transitive group 9T12
Generators in S9
(1 2 3)(4 5 6)(7 8 9)
(1 9 5)(2 7 6)(3 8 4)
(2 7 6)(3 4 8)
(2 3)(4 6)(7 8)

G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4), (2,7,6)(3,4,8), (2,3)(4,6)(7,8)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4), (2,7,6)(3,4,8), (2,3)(4,6)(7,8) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9)], [(1,9,5),(2,7,6),(3,8,4)], [(2,7,6),(3,4,8)], [(2,3),(4,6),(7,8)])

G:=TransitiveGroup(9,12);

On 18 points - transitive group 18T24
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 11 13)(2 12 14)(3 10 15)(4 9 18)(5 7 16)(6 8 17)
(2 12 14)(3 15 10)(4 9 18)(5 16 7)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 17)(14 16)(15 18)

G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11,13)(2,12,14)(3,10,15)(4,9,18)(5,7,16)(6,8,17), (2,12,14)(3,15,10)(4,9,18)(5,16,7), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,17)(14,16)(15,18)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11,13)(2,12,14)(3,10,15)(4,9,18)(5,7,16)(6,8,17), (2,12,14)(3,15,10)(4,9,18)(5,16,7), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,17)(14,16)(15,18) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,11,13),(2,12,14),(3,10,15),(4,9,18),(5,7,16),(6,8,17)], [(2,12,14),(3,15,10),(4,9,18),(5,16,7)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,17),(14,16),(15,18)])

G:=TransitiveGroup(18,24);

On 27 points - transitive group 27T6
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 13 11)(2 14 12)(3 15 10)(4 26 8)(5 27 9)(6 25 7)(16 19 22)(17 20 23)(18 21 24)
(1 5 24)(2 25 19)(3 8 17)(4 20 15)(6 16 12)(7 22 14)(9 21 11)(10 26 23)(13 27 18)
(2 3)(4 22)(5 24)(6 23)(7 20)(8 19)(9 21)(10 12)(14 15)(16 26)(17 25)(18 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), (1,13,11)(2,14,12)(3,15,10)(4,26,8)(5,27,9)(6,25,7)(16,19,22)(17,20,23)(18,21,24), (1,5,24)(2,25,19)(3,8,17)(4,20,15)(6,16,12)(7,22,14)(9,21,11)(10,26,23)(13,27,18), (2,3)(4,22)(5,24)(6,23)(7,20)(8,19)(9,21)(10,12)(14,15)(16,26)(17,25)(18,27)>;

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

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

G:=TransitiveGroup(27,6);

He3⋊C2 is a maximal subgroup of
He3⋊C4  C32⋊D6  C3≀S3  He3.C6  He3.2C6  He3.4C6  He35S3  C32⋊S4  C32⋊D15  C32⋊D21
He3⋊C2 is a maximal quotient of
He33C4  C322D9  C33⋊S3  He3.3S3  He3⋊S3  3- 1+2.S3  He35S3  C32⋊S4  C32⋊D15  C32⋊D21

Polynomial with Galois group He3⋊C2 over ℚ
actionf(x)Disc(f)
9T12x9+x8-54x7+68x6+695x5-1857x4-473x3+6301x2-7401x+2727212·325·114·472·1073

Matrix representation of He3⋊C2 in GL3(𝔽7) generated by

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

He3⋊C2 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_2
% in TeX

G:=Group("He3:C2");
// GroupNames label

G:=SmallGroup(54,8);
// by ID

G=gap.SmallGroup(54,8);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,33,146,150]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

׿
×
𝔽