metacyclic, supersoluble, monomial
Aliases: C8⋊2F7, C56⋊2C6, D28.2C6, Dic14⋊4C6, C56⋊C2⋊C3, C7⋊C3⋊1SD16, C7⋊1(C3×SD16), C4⋊F7.2C2, C4.F7⋊4C2, C4.8(C2×F7), C28.8(C2×C6), C14.1(C3×D4), C2.3(C4⋊F7), (C8×C7⋊C3)⋊2C2, (C2×C7⋊C3).1D4, (C4×C7⋊C3).8C22, SmallGroup(336,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊C6
G = < a,b | a56=b6=1, bab-1=a3 >
Character table of C56⋊C6
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 7 | 8A | 8B | 12A | 12B | 12C | 12D | 14 | 24A | 24B | 24C | 24D | 28A | 28B | 56A | 56B | 56C | 56D | |
| size | 1 | 1 | 28 | 7 | 7 | 2 | 28 | 7 | 7 | 28 | 28 | 6 | 2 | 2 | 14 | 14 | 28 | 28 | 6 | 14 | 14 | 14 | 14 | 6 | 6 | 6 | 6 | 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 | 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 | -1 | linear of order 2 |
| ρ3 | 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 | 1 | linear of order 2 |
| ρ4 | 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 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ13 | 2 | 2 | 0 | 2 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | -2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 2 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ15 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | -2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 2 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ16 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | -2 | -√-2 | √-2 | √-2 | -√-2 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ17 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | -2 | √-2 | -√-2 | -√-2 | √-2 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ18 | 2 | -2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | -2 | ζ83ζ32+ζ8ζ32 | ζ87ζ3+ζ85ζ3 | ζ87ζ32+ζ85ζ32 | ζ83ζ3+ζ8ζ3 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from C3×SD16 |
| ρ19 | 2 | -2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | -2 | ζ83ζ3+ζ8ζ3 | ζ87ζ32+ζ85ζ32 | ζ87ζ3+ζ85ζ3 | ζ83ζ32+ζ8ζ32 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from C3×SD16 |
| ρ20 | 2 | -2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | -2 | ζ87ζ32+ζ85ζ32 | ζ83ζ3+ζ8ζ3 | ζ83ζ32+ζ8ζ32 | ζ87ζ3+ζ85ζ3 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from C3×SD16 |
| ρ21 | 2 | -2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | -2 | ζ87ζ3+ζ85ζ3 | ζ83ζ32+ζ8ζ32 | ζ83ζ3+ζ8ζ3 | ζ87ζ32+ζ85ζ32 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from C3×SD16 |
| ρ22 | 6 | 6 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | 6 | 6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ23 | 6 | 6 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | -6 | -6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F7 |
| ρ24 | 6 | 6 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -√7 | -√7 | √7 | √7 | orthogonal lifted from C4⋊F7 |
| ρ25 | 6 | 6 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | √7 | √7 | -√7 | -√7 | orthogonal lifted from C4⋊F7 |
| ρ26 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -3√-2 | 3√-2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -√7 | √7 | ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ83-ζ8ζ74-ζ8ζ72-ζ8ζ7 | ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ87-ζ85ζ74-ζ85ζ72-ζ85ζ7 | ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ87-ζ85ζ76-ζ85ζ75-ζ85ζ73 | -ζ83ζ74-ζ83ζ72-ζ83ζ7+ζ8ζ74+ζ8ζ72+ζ8ζ7+ζ8 | complex faithful |
| ρ27 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3√-2 | -3√-2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | √7 | -√7 | ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ87-ζ85ζ76-ζ85ζ75-ζ85ζ73 | -ζ83ζ74-ζ83ζ72-ζ83ζ7+ζ8ζ74+ζ8ζ72+ζ8ζ7+ζ8 | ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ83-ζ8ζ74-ζ8ζ72-ζ8ζ7 | ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ87-ζ85ζ74-ζ85ζ72-ζ85ζ7 | complex faithful |
| ρ28 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3√-2 | -3√-2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -√7 | √7 | ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ87-ζ85ζ74-ζ85ζ72-ζ85ζ7 | ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ83-ζ8ζ74-ζ8ζ72-ζ8ζ7 | -ζ83ζ74-ζ83ζ72-ζ83ζ7+ζ8ζ74+ζ8ζ72+ζ8ζ7+ζ8 | ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ87-ζ85ζ76-ζ85ζ75-ζ85ζ73 | complex faithful |
| ρ29 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -3√-2 | 3√-2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | √7 | -√7 | -ζ83ζ74-ζ83ζ72-ζ83ζ7+ζ8ζ74+ζ8ζ72+ζ8ζ7+ζ8 | ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ87-ζ85ζ76-ζ85ζ75-ζ85ζ73 | ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ87-ζ85ζ74-ζ85ζ72-ζ85ζ7 | ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ83-ζ8ζ74-ζ8ζ72-ζ8ζ7 | complex faithful |
►On 56 points
Generators in S56
(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)
(2 20 26 28 10 4)(3 39 51 55 19 7)(5 21 45 53 37 13)(6 40 14 24 46 16)(8 22)(9 41 33 49 17 25)(11 23 27 47 35 31)(12 42 52 18 44 34)(15 43)(30 48 54 56 38 32)(36 50)
G:=sub<Sym(56)| (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), (2,20,26,28,10,4)(3,39,51,55,19,7)(5,21,45,53,37,13)(6,40,14,24,46,16)(8,22)(9,41,33,49,17,25)(11,23,27,47,35,31)(12,42,52,18,44,34)(15,43)(30,48,54,56,38,32)(36,50)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56), (2,20,26,28,10,4)(3,39,51,55,19,7)(5,21,45,53,37,13)(6,40,14,24,46,16)(8,22)(9,41,33,49,17,25)(11,23,27,47,35,31)(12,42,52,18,44,34)(15,43)(30,48,54,56,38,32)(36,50) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)], [(2,20,26,28,10,4),(3,39,51,55,19,7),(5,21,45,53,37,13),(6,40,14,24,46,16),(8,22),(9,41,33,49,17,25),(11,23,27,47,35,31),(12,42,52,18,44,34),(15,43),(30,48,54,56,38,32),(36,50)]])
Matrix representation of C56⋊C6 ►in GL6(𝔽3)
| 0 | 1 | 2 | 0 | 0 | 2 |
| 1 | 2 | 1 | 2 | 0 | 1 |
| 0 | 2 | 2 | 2 | 0 | 0 |
| 0 | 1 | 2 | 2 | 1 | 2 |
| 0 | 1 | 0 | 1 | 0 | 2 |
| 0 | 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 2 | 0 | 2 |
| 0 | 1 | 1 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 2 | 0 | 2 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 2 |
G:=sub<GL(6,GF(3))| [0,1,0,0,0,0,1,2,2,1,1,0,2,1,2,2,0,1,0,2,2,2,1,1,0,0,0,1,0,0,2,1,0,2,2,1],[1,0,0,0,0,0,0,1,1,0,2,1,0,1,0,0,0,0,2,2,0,2,2,1,0,0,1,0,0,0,2,0,0,0,0,2] >;
C56⋊C6 in GAP, Magma, Sage, TeX
C_{56}\rtimes C_6 % in TeX
G:=Group("C56:C6"); // GroupNames label
G:=SmallGroup(336,9);
// by ID
G=gap.SmallGroup(336,9);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,79,867,69,10373,1745]);
// Polycyclic
G:=Group<a,b|a^56=b^6=1,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C56⋊C6 in TeX
Character table of C56⋊C6 in TeX