non-abelian, supersoluble, monomial
Aliases: He3⋊3C4, C32⋊2Dic3, (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
| He3 — He3⋊3C4 |
Generators and relations for He3⋊3C4
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 >
Character table of He3⋊3C4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ10 | 2 | -2 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | -2 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | -2 | -2 | 1 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ3 | ζ32 | complex lifted from He3⋊C2 |
| ρ14 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from He3⋊C2 |
| ρ15 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ32 | ζ3 | complex lifted from He3⋊C2 |
| ρ16 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from He3⋊C2 |
| ρ17 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -i | i | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | complex faithful |
| ρ18 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | i | -i | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | complex faithful |
| ρ19 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -i | i | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | complex faithful |
| ρ20 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | i | -i | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | complex faithful |
►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)]])
He3⋊3C4 is a maximal subgroup of
He3⋊2C8 He3⋊2Q8 C6.S32 He3⋊2D4 He3⋊4Q8 C4×He3⋊C2 He3⋊7D4 He3⋊C12 He3.C12 He3.2C12 He3.5C12 He3⋊6Dic3 C32⋊2CSU2(𝔽3) C62⋊6Dic3
He3⋊3C4 is a maximal quotient of
He3⋊4C8 C32⋊2Dic9 C33⋊Dic3 He3.3Dic3 He3⋊Dic3 3- 1+2.Dic3 He3⋊6Dic3 C62⋊6Dic3
Matrix representation of He3⋊3C4 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 9 |
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 0 | 8 | 0 |
| 0 | 0 | 10 |
| 7 | 0 | 0 |
| 8 | 0 | 0 |
| 0 | 0 | 2 |
| 0 | 6 | 0 |
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] >;
He3⋊3C4 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 He3⋊3C4 in TeX
Character table of He3⋊3C4 in TeX