Copied to
clipboard

G = He33C4order 108 = 22·33

2nd semidirect product of He3 and C4 acting via C4/C2=C2

non-abelian, supersoluble, monomial

Aliases: He33C4, C322Dic3, (C3×C6).3S3, C6.4(C3⋊S3), C2.(He3⋊C2), (C2×He3).2C2, C3.2(C3⋊Dic3), SmallGroup(108,11)

Series: Derived Chief Lower central Upper central

C1C3He3 — He33C4
C1C3C32He3C2×He3 — He33C4
He3 — He33C4
C1C6

Generators and relations for He33C4
 G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

3C3
3C3
3C3
3C3
9C4
3C6
3C6
3C6
3C6
3Dic3
3Dic3
3Dic3
3Dic3
9C12
3C3×Dic3
3C3×Dic3
3C3×Dic3
3C3×Dic3

Character table of He33C4

 class 123A3B3C3D3E3F4A4B6A6B6C6D6E6F12A12B12C12D
 size 11116666991166669999
ρ111111111111111111111    trivial
ρ211111111-1-1111111-1-1-1-1    linear of order 2
ρ31-1111111i-i-1-1-1-1-1-1-i-iii    linear of order 4
ρ41-1111111-ii-1-1-1-1-1-1ii-i-i    linear of order 4
ρ52222-1-1-1200222-1-1-10000    orthogonal lifted from S3
ρ62222-12-1-10022-1-1-120000    orthogonal lifted from S3
ρ722222-1-1-10022-1-12-10000    orthogonal lifted from S3
ρ82222-1-12-10022-12-1-10000    orthogonal lifted from S3
ρ92-222-1-1-1200-2-2-21110000    symplectic lifted from Dic3, Schur index 2
ρ102-222-12-1-100-2-2111-20000    symplectic lifted from Dic3, Schur index 2
ρ112-2222-1-1-100-2-211-210000    symplectic lifted from Dic3, Schur index 2
ρ122-222-1-12-100-2-21-2110000    symplectic lifted from Dic3, Schur index 2
ρ1333-3-3-3/2-3+3-3/2000011-3+3-3/2-3-3-3/20000ζ32ζ3ζ3ζ32    complex lifted from He3⋊C2
ρ1433-3+3-3/2-3-3-3/20000-1-1-3-3-3/2-3+3-3/20000ζ65ζ6ζ6ζ65    complex lifted from He3⋊C2
ρ1533-3+3-3/2-3-3-3/2000011-3-3-3/2-3+3-3/20000ζ3ζ32ζ32ζ3    complex lifted from He3⋊C2
ρ1633-3-3-3/2-3+3-3/20000-1-1-3+3-3/2-3-3-3/20000ζ6ζ65ζ65ζ6    complex lifted from He3⋊C2
ρ173-3-3+3-3/2-3-3-3/20000-ii3+3-3/23-3-3/20000ζ4ζ3ζ4ζ32ζ43ζ32ζ43ζ3    complex faithful
ρ183-3-3-3-3/2-3+3-3/20000i-i3-3-3/23+3-3/20000ζ43ζ32ζ43ζ3ζ4ζ3ζ4ζ32    complex faithful
ρ193-3-3-3-3/2-3+3-3/20000-ii3-3-3/23+3-3/20000ζ4ζ32ζ4ζ3ζ43ζ3ζ43ζ32    complex faithful
ρ203-3-3+3-3/2-3-3-3/20000i-i3+3-3/23-3-3/20000ζ43ζ3ζ43ζ32ζ4ζ32ζ4ζ3    complex faithful

Smallest permutation representation of He33C4
On 36 points
Generators in S36
(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(1 16 33)(2 34 13)(3 14 35)(4 36 15)(5 31 20)(6 17 32)(7 29 18)(8 19 30)(9 21 27)(10 28 22)(11 23 25)(12 26 24)
(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 28)(29 30 31 32)(33 34 35 36)

G:=sub<Sym(36)| (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,16,33)(2,34,13)(3,14,35)(4,36,15)(5,31,20)(6,17,32)(7,29,18)(8,19,30)(9,21,27)(10,28,22)(11,23,25)(12,26,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)>;

G:=Group( (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,16,33)(2,34,13)(3,14,35)(4,36,15)(5,31,20)(6,17,32)(7,29,18)(8,19,30)(9,21,27)(10,28,22)(11,23,25)(12,26,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36) );

G=PermutationGroup([(1,16,7),(2,8,13),(3,14,5),(4,6,15),(9,29,18),(10,19,30),(11,31,20),(12,17,32),(21,27,33),(22,34,28),(23,25,35),(24,36,26)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,11),(6,36,12),(7,33,9),(8,34,10),(13,28,19),(14,25,20),(15,26,17),(16,27,18)], [(1,16,33),(2,34,13),(3,14,35),(4,36,15),(5,31,20),(6,17,32),(7,29,18),(8,19,30),(9,21,27),(10,28,22),(11,23,25),(12,26,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)])

He33C4 is a maximal subgroup of
He32C8  He32Q8  C6.S32  He32D4  He34Q8  C4×He3⋊C2  He37D4  He3⋊C12  He3.C12  He3.2C12  He3.5C12  He36Dic3  C322CSU2(𝔽3)  C626Dic3
He33C4 is a maximal quotient of
He34C8  C322Dic9  C33⋊Dic3  He3.3Dic3  He3⋊Dic3  3- 1+2.Dic3  He36Dic3  C626Dic3

Matrix representation of He33C4 in GL3(𝔽13) generated by

100
030
009
,
300
030
003
,
080
0010
700
,
800
002
060
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,9],[3,0,0,0,3,0,0,0,3],[0,0,7,8,0,0,0,10,0],[8,0,0,0,0,6,0,2,0] >;

He33C4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3C_4
% in TeX

G:=Group("He3:3C4");
// GroupNames label

G:=SmallGroup(108,11);
// by ID

G=gap.SmallGroup(108,11);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,10,122,483,253]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^4=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of He33C4 in TeX
Character table of He33C4 in TeX

׿
×
𝔽