metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2S3, C24⋊2C2, C6.1D4, C4.8D6, C3⋊1SD16, C2.3D12, Dic6⋊1C2, D12.1C2, C12.8C22, SmallGroup(48,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊C2
G = < a,b | a24=b2=1, bab=a11 >
Character table of C24⋊C2
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 12 | 2 | 2 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | -1 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | -1 | 2 | 0 | -1 | -2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | 1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ9 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | 1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ10 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -√-2 | √-2 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ11 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | √-2 | -√-2 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ12 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | -√-2 | √-2 | -√3 | √3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | complex faithful |
| ρ13 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | √-2 | -√-2 | -√3 | √3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | complex faithful |
| ρ14 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | -√-2 | √-2 | √3 | -√3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | complex faithful |
| ρ15 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | √-2 | -√-2 | √3 | -√3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | complex faithful |
►On 24 points - transitive group 24T35
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)
G:=sub<Sym(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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)>;
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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20) );
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)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20)]])
G:=TransitiveGroup(24,35);
C24⋊C2 is a maximal subgroup of
C4○D24 C8⋊D6 C8.D6 D8⋊S3 S3×SD16 Q8.7D6 Q16⋊S3 C72⋊C2 D12.S3 C32⋊5SD16 C24⋊2S3 C8⋊2S4 C8.4S4 D12.D5 Dic6⋊D5 C24⋊D5 D12.D7 Dic6⋊D7 C8⋊D21 He3⋊2SD16 C33⋊8SD16 F9⋊S3
C24⋊C2 is a maximal quotient of
C2.Dic12 C8⋊Dic3 C2.D24 C72⋊C2 D12.S3 C32⋊5SD16 C24⋊2S3 C8⋊2S4 D12.D5 Dic6⋊D5 C24⋊D5 D12.D7 Dic6⋊D7 C8⋊D21 C33⋊8SD16 F9⋊S3
Matrix representation of C24⋊C2 ►in GL2(𝔽11) generated by
| 0 | 1 |
| 1 | 5 |
| 7 | 5 |
| 8 | 4 |
G:=sub<GL(2,GF(11))| [0,1,1,5],[7,8,5,4] >;
C24⋊C2 in GAP, Magma, Sage, TeX
C_{24}\rtimes C_2 % in TeX
G:=Group("C24:C2"); // GroupNames label
G:=SmallGroup(48,6);
// by ID
G=gap.SmallGroup(48,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,61,26,182,42,804]);
// Polycyclic
G:=Group<a,b|a^24=b^2=1,b*a*b=a^11>;
// generators/relations
Export
Subgroup lattice of C24⋊C2 in TeX
Character table of C24⋊C2 in TeX