Aliases: He3⋊C4, C3.(C32⋊C4), He3⋊C2.C2, SmallGroup(108,15)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3⋊C4 |
Generators and relations for He3⋊C4
G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, cac-1=ab-1, dad-1=abc, bc=cb, bd=db, dcd-1=ac-1 >
Character table of He3⋊C4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 9 | 1 | 1 | 12 | 12 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ5 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 1 | 1 | ζ6 | ζ65 | ζ32 | ζ3 | ζ3 | ζ32 | complex faithful |
| ρ6 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 1 | 1 | ζ65 | ζ6 | ζ3 | ζ32 | ζ32 | ζ3 | complex faithful |
| ρ7 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | complex faithful |
| ρ8 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | complex faithful |
| ρ9 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | i | -i | ζ3 | ζ32 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | complex faithful |
| ρ10 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -i | i | ζ32 | ζ3 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | complex faithful |
| ρ11 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | i | -i | ζ32 | ζ3 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | complex faithful |
| ρ12 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -i | i | ζ3 | ζ32 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | complex faithful |
| ρ13 | 4 | 0 | 4 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ14 | 4 | 0 | 4 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
►On 18 points - transitive group 18T49
Generators in S18
(1 8 18)(2 5 3)(4 11 10)(6 16 13)(7 14 15)
(1 4 6)(2 3 5)(7 14 15)(8 11 16)(9 12 17)(10 13 18)
(1 18 16)(2 7 9)(3 14 12)(4 10 8)(5 15 17)(6 13 11)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
G:=sub<Sym(18)| (1,8,18)(2,5,3)(4,11,10)(6,16,13)(7,14,15), (1,4,6)(2,3,5)(7,14,15)(8,11,16)(9,12,17)(10,13,18), (1,18,16)(2,7,9)(3,14,12)(4,10,8)(5,15,17)(6,13,11), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;
G:=Group( (1,8,18)(2,5,3)(4,11,10)(6,16,13)(7,14,15), (1,4,6)(2,3,5)(7,14,15)(8,11,16)(9,12,17)(10,13,18), (1,18,16)(2,7,9)(3,14,12)(4,10,8)(5,15,17)(6,13,11), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );
G=PermutationGroup([[(1,8,18),(2,5,3),(4,11,10),(6,16,13),(7,14,15)], [(1,4,6),(2,3,5),(7,14,15),(8,11,16),(9,12,17),(10,13,18)], [(1,18,16),(2,7,9),(3,14,12),(4,10,8),(5,15,17),(6,13,11)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)]])
G:=TransitiveGroup(18,49);
►On 27 points - transitive group 27T32
Generators in S27
(1 8 4)(2 6 13)(3 15 10)(5 17 16)(7 23 26)(9 25 19)(11 22 21)(12 18 20)(14 24 27)
(1 3 2)(4 10 13)(5 11 14)(6 8 15)(7 9 12)(16 21 27)(17 22 24)(18 23 25)(19 20 26)
(1 27 25)(2 21 23)(3 16 18)(4 22 12)(5 26 8)(6 14 20)(7 10 24)(9 13 17)(11 19 15)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
G:=sub<Sym(27)| (1,8,4)(2,6,13)(3,15,10)(5,17,16)(7,23,26)(9,25,19)(11,22,21)(12,18,20)(14,24,27), (1,3,2)(4,10,13)(5,11,14)(6,8,15)(7,9,12)(16,21,27)(17,22,24)(18,23,25)(19,20,26), (1,27,25)(2,21,23)(3,16,18)(4,22,12)(5,26,8)(6,14,20)(7,10,24)(9,13,17)(11,19,15), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27)>;
G:=Group( (1,8,4)(2,6,13)(3,15,10)(5,17,16)(7,23,26)(9,25,19)(11,22,21)(12,18,20)(14,24,27), (1,3,2)(4,10,13)(5,11,14)(6,8,15)(7,9,12)(16,21,27)(17,22,24)(18,23,25)(19,20,26), (1,27,25)(2,21,23)(3,16,18)(4,22,12)(5,26,8)(6,14,20)(7,10,24)(9,13,17)(11,19,15), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27) );
G=PermutationGroup([[(1,8,4),(2,6,13),(3,15,10),(5,17,16),(7,23,26),(9,25,19),(11,22,21),(12,18,20),(14,24,27)], [(1,3,2),(4,10,13),(5,11,14),(6,8,15),(7,9,12),(16,21,27),(17,22,24),(18,23,25),(19,20,26)], [(1,27,25),(2,21,23),(3,16,18),(4,22,12),(5,26,8),(6,14,20),(7,10,24),(9,13,17),(11,19,15)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)]])
G:=TransitiveGroup(27,32);
He3⋊C4 is a maximal subgroup of
He3⋊C8 He3⋊D4 SU3(𝔽2) He3.3C12 He3⋊4Dic3
He3⋊C4 is a maximal quotient of He3⋊2C8 He3⋊4Dic3
Matrix representation of He3⋊C4 ►in GL3(𝔽7) generated by
| 0 | 5 | 1 |
| 0 | 2 | 3 |
| 6 | 6 | 5 |
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 0 | 0 | 1 |
| 0 | 2 | 3 |
| 3 | 0 | 5 |
| 0 | 1 | 3 |
| 4 | 3 | 5 |
| 0 | 5 | 3 |
G:=sub<GL(3,GF(7))| [0,0,6,5,2,6,1,3,5],[2,0,0,0,2,0,0,0,2],[0,0,3,0,2,0,1,3,5],[0,4,0,1,3,5,3,5,3] >;
He3⋊C4 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes C_4 % in TeX
G:=Group("He3:C4"); // GroupNames label
G:=SmallGroup(108,15);
// by ID
G=gap.SmallGroup(108,15);
# by ID
G:=PCGroup([5,-2,-2,-3,3,-3,10,422,67,643,608,253]);
// 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*b*c,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations
Export
Subgroup lattice of He3⋊C4 in TeX
Character table of He3⋊C4 in TeX