metacyclic, supersoluble, monomial, Z-group
Aliases: C7⋊C12, C2.F7, C14.C6, Dic7⋊C3, C7⋊C3⋊C4, (C2×C7⋊C3).C2, SmallGroup(84,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C12 |
Generators and relations for C7⋊C12
G = < a,b | a7=b12=1, bab-1=a5 >
Character table of C7⋊C12
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7 | 12A | 12B | 12C | 12D | 14 | |
| size | 1 | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 7 | 7 | 7 | 7 | 6 | |
| 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 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | 1 | -i | -i | i | i | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | 1 | i | i | -i | -i | -1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ65 | ζ6 | 1 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | -1 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ65 | ζ6 | 1 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | -1 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ6 | ζ65 | 1 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | -1 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ6 | ζ65 | 1 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | -1 | linear of order 12 |
| ρ13 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from F7 |
| ρ14 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | symplectic faithful, Schur index 2 |
►On 28 points - transitive group 28T12
Generators in S28
(1 5 13 26 9 18 22)(2 19 27 6 23 10 14)(3 11 7 20 15 24 28)(4 25 21 12 17 16 8)
(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)
G:=sub<Sym(28)| (1,5,13,26,9,18,22)(2,19,27,6,23,10,14)(3,11,7,20,15,24,28)(4,25,21,12,17,16,8), (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)>;
G:=Group( (1,5,13,26,9,18,22)(2,19,27,6,23,10,14)(3,11,7,20,15,24,28)(4,25,21,12,17,16,8), (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) );
G=PermutationGroup([[(1,5,13,26,9,18,22),(2,19,27,6,23,10,14),(3,11,7,20,15,24,28),(4,25,21,12,17,16,8)], [(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)]])
G:=TransitiveGroup(28,12);
C7⋊C12 is a maximal subgroup of
C4.F7 C4×F7 Dic7⋊C6 C6.F7 Q8.F7 Dic7⋊A4 C35⋊3C12 C35⋊C12
C7⋊C12 is a maximal quotient of
C7⋊C24 C7⋊C36 C6.F7 Dic7⋊A4 C35⋊3C12 C35⋊C12
Matrix representation of C7⋊C12 ►in GL6(𝔽3)
| 0 | 1 | 2 | 0 | 0 | 0 |
| 2 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 2 |
| 0 | 1 | 2 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 2 | 2 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 2 | 0 | 0 | 2 | 0 | 1 |
| 1 | 1 | 2 | 0 | 0 | 2 |
G:=sub<GL(6,GF(3))| [0,2,0,0,0,0,1,0,0,1,0,2,2,0,0,2,1,2,0,1,2,2,0,0,0,0,0,0,0,1,0,0,2,0,0,0],[1,1,1,0,2,1,0,0,0,0,0,1,1,0,0,1,0,2,0,0,0,0,2,0,0,0,0,1,0,0,1,1,0,1,1,2] >;
C7⋊C12 in GAP, Magma, Sage, TeX
C_7\rtimes C_{12} % in TeX
G:=Group("C7:C12"); // GroupNames label
G:=SmallGroup(84,1);
// by ID
G=gap.SmallGroup(84,1);
# by ID
G:=PCGroup([4,-2,-3,-2,-7,24,1155,391]);
// Polycyclic
G:=Group<a,b|a^7=b^12=1,b*a*b^-1=a^5>;
// generators/relations
Export
Subgroup lattice of C7⋊C12 in TeX
Character table of C7⋊C12 in TeX