metacyclic, supersoluble, monomial, Z-group
Aliases: C7⋊C24, C14.C12, C4.2F7, C28.2C6, C7⋊C8⋊C3, C7⋊C3⋊C8, C2.(C7⋊C12), (C2×C7⋊C3).C4, (C4×C7⋊C3).2C2, SmallGroup(168,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C24 |
Generators and relations for C7⋊C24
G = < a,b | a7=b24=1, bab-1=a3 >
Character table of C7⋊C24
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7 | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 14 | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 28A | 28B | |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | -1 | 1 | -i | i | i | -i | -i | i | i | -i | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | 1 | ζ8 | ζ87 | ζ83 | ζ85 | -i | -i | i | i | -1 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | i | -i | linear of order 8 |
| ρ10 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | 1 | ζ83 | ζ85 | ζ8 | ζ87 | i | i | -i | -i | -1 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | -i | i | linear of order 8 |
| ρ11 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | 1 | ζ87 | ζ8 | ζ85 | ζ83 | i | i | -i | -i | -1 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | -i | i | linear of order 8 |
| ρ12 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | 1 | ζ85 | ζ83 | ζ87 | ζ8 | -i | -i | i | i | -1 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | i | -i | linear of order 8 |
| ρ13 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | 1 | -i | i | i | -i | ζ6 | ζ65 | ζ6 | ζ65 | 1 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | -1 | -1 | linear of order 12 |
| ρ14 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | 1 | i | -i | -i | i | ζ6 | ζ65 | ζ6 | ζ65 | 1 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | -1 | -1 | linear of order 12 |
| ρ15 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | 1 | -i | i | i | -i | ζ65 | ζ6 | ζ65 | ζ6 | 1 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | -1 | -1 | linear of order 12 |
| ρ16 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | 1 | i | -i | -i | i | ζ65 | ζ6 | ζ65 | ζ6 | 1 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | -1 | -1 | linear of order 12 |
| ρ17 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ6 | ζ65 | 1 | ζ87 | ζ8 | ζ85 | ζ83 | ζ82ζ3 | ζ82ζ32 | ζ86ζ3 | ζ86ζ32 | -1 | ζ87ζ3 | ζ8ζ32 | ζ85ζ32 | ζ83ζ32 | ζ87ζ32 | ζ8ζ3 | ζ85ζ3 | ζ83ζ3 | -i | i | linear of order 24 |
| ρ18 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ65 | ζ6 | 1 | ζ83 | ζ85 | ζ8 | ζ87 | ζ82ζ32 | ζ82ζ3 | ζ86ζ32 | ζ86ζ3 | -1 | ζ83ζ32 | ζ85ζ3 | ζ8ζ3 | ζ87ζ3 | ζ83ζ3 | ζ85ζ32 | ζ8ζ32 | ζ87ζ32 | -i | i | linear of order 24 |
| ρ19 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ6 | ζ65 | 1 | ζ8 | ζ87 | ζ83 | ζ85 | ζ86ζ3 | ζ86ζ32 | ζ82ζ3 | ζ82ζ32 | -1 | ζ8ζ3 | ζ87ζ32 | ζ83ζ32 | ζ85ζ32 | ζ8ζ32 | ζ87ζ3 | ζ83ζ3 | ζ85ζ3 | i | -i | linear of order 24 |
| ρ20 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ65 | ζ6 | 1 | ζ87 | ζ8 | ζ85 | ζ83 | ζ82ζ32 | ζ82ζ3 | ζ86ζ32 | ζ86ζ3 | -1 | ζ87ζ32 | ζ8ζ3 | ζ85ζ3 | ζ83ζ3 | ζ87ζ3 | ζ8ζ32 | ζ85ζ32 | ζ83ζ32 | -i | i | linear of order 24 |
| ρ21 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ65 | ζ6 | 1 | ζ85 | ζ83 | ζ87 | ζ8 | ζ86ζ32 | ζ86ζ3 | ζ82ζ32 | ζ82ζ3 | -1 | ζ85ζ32 | ζ83ζ3 | ζ87ζ3 | ζ8ζ3 | ζ85ζ3 | ζ83ζ32 | ζ87ζ32 | ζ8ζ32 | i | -i | linear of order 24 |
| ρ22 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ6 | ζ65 | 1 | ζ83 | ζ85 | ζ8 | ζ87 | ζ82ζ3 | ζ82ζ32 | ζ86ζ3 | ζ86ζ32 | -1 | ζ83ζ3 | ζ85ζ32 | ζ8ζ32 | ζ87ζ32 | ζ83ζ32 | ζ85ζ3 | ζ8ζ3 | ζ87ζ3 | -i | i | linear of order 24 |
| ρ23 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ65 | ζ6 | 1 | ζ8 | ζ87 | ζ83 | ζ85 | ζ86ζ32 | ζ86ζ3 | ζ82ζ32 | ζ82ζ3 | -1 | ζ8ζ32 | ζ87ζ3 | ζ83ζ3 | ζ85ζ3 | ζ8ζ3 | ζ87ζ32 | ζ83ζ32 | ζ85ζ32 | i | -i | linear of order 24 |
| ρ24 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ6 | ζ65 | 1 | ζ85 | ζ83 | ζ87 | ζ8 | ζ86ζ3 | ζ86ζ32 | ζ82ζ3 | ζ82ζ32 | -1 | ζ85ζ3 | ζ83ζ32 | ζ87ζ32 | ζ8ζ32 | ζ85ζ32 | ζ83ζ3 | ζ87ζ3 | ζ8ζ3 | i | -i | linear of order 24 |
| ρ25 | 6 | 6 | 0 | 0 | 6 | 6 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F7 |
| ρ26 | 6 | 6 | 0 | 0 | -6 | -6 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from C7⋊C12, Schur index 2 |
| ρ27 | 6 | -6 | 0 | 0 | 6i | -6i | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | complex faithful |
| ρ28 | 6 | -6 | 0 | 0 | -6i | 6i | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | complex faithful |
►On 56 points
Generators in S56
(1 17 9 56 25 40 48)(2 33 49 10 41 18 26)(3 11 27 50 19 34 42)(4 51 43 28 35 12 20)(5 29 21 44 13 52 36)(6 45 37 22 53 30 14)(7 23 15 38 31 46 54)(8 39 55 16 47 24 32)
(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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)
G:=sub<Sym(56)| (1,17,9,56,25,40,48)(2,33,49,10,41,18,26)(3,11,27,50,19,34,42)(4,51,43,28,35,12,20)(5,29,21,44,13,52,36)(6,45,37,22,53,30,14)(7,23,15,38,31,46,54)(8,39,55,16,47,24,32), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)>;
G:=Group( (1,17,9,56,25,40,48)(2,33,49,10,41,18,26)(3,11,27,50,19,34,42)(4,51,43,28,35,12,20)(5,29,21,44,13,52,36)(6,45,37,22,53,30,14)(7,23,15,38,31,46,54)(8,39,55,16,47,24,32), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56) );
G=PermutationGroup([[(1,17,9,56,25,40,48),(2,33,49,10,41,18,26),(3,11,27,50,19,34,42),(4,51,43,28,35,12,20),(5,29,21,44,13,52,36),(6,45,37,22,53,30,14),(7,23,15,38,31,46,54),(8,39,55,16,47,24,32)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)]])
C7⋊C24 is a maximal subgroup of
C8×F7 C8⋊F7 C28.C12 D4⋊F7 D4.F7 Q8⋊2F7 Q8.2F7
C7⋊C24 is a maximal quotient of C7⋊C48
Matrix representation of C7⋊C24 ►in GL7(𝔽337)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 336 | 1 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 1 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 1 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 | 1 | 0 |
| 0 | 336 | 0 | 0 | 0 | 0 | 1 |
| 0 | 336 | 0 | 0 | 0 | 0 | 0 |
| 241 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 134 | 134 | 254 | 0 | 203 | 0 |
| 0 | 0 | 134 | 203 | 134 | 0 | 254 |
| 0 | 134 | 51 | 203 | 0 | 0 | 203 |
| 0 | 134 | 0 | 0 | 134 | 254 | 203 |
| 0 | 51 | 0 | 203 | 134 | 203 | 0 |
| 0 | 0 | 134 | 0 | 51 | 203 | 203 |
G:=sub<GL(7,GF(337))| [1,0,0,0,0,0,0,0,336,336,336,336,336,336,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[241,0,0,0,0,0,0,0,134,0,134,134,51,0,0,134,134,51,0,0,134,0,254,203,203,0,203,0,0,0,134,0,134,134,51,0,203,0,0,254,203,203,0,0,254,203,203,0,203] >;
C7⋊C24 in GAP, Magma, Sage, TeX
C_7\rtimes C_{24} % in TeX
G:=Group("C7:C24"); // GroupNames label
G:=SmallGroup(168,1);
// by ID
G=gap.SmallGroup(168,1);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-7,30,42,3604,1209]);
// Polycyclic
G:=Group<a,b|a^7=b^24=1,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C7⋊C24 in TeX
Character table of C7⋊C24 in TeX