metacyclic, supersoluble, monomial
Aliases: C4⋊F7, D28⋊C3, C28⋊1C6, D14⋊1C6, C7⋊C3⋊1D4, C7⋊1(C3×D4), (C2×F7)⋊1C2, C2.4(C2×F7), C14.3(C2×C6), (C4×C7⋊C3)⋊1C2, (C2×C7⋊C3).3C22, SmallGroup(168,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊F7
G = < a,b,c | a4=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >
Character table of C4⋊F7
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7 | 12A | 12B | 14 | 28A | 28B | |
| size | 1 | 1 | 14 | 14 | 7 | 7 | 2 | 7 | 7 | 14 | 14 | 14 | 14 | 6 | 14 | 14 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | 1 | ζ65 | ζ6 | 1 | -1 | -1 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | 1 | ζ6 | ζ65 | 1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ65 | ζ32 | ζ6 | ζ3 | 1 | ζ65 | ζ6 | 1 | -1 | -1 | linear of order 6 |
| ρ10 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ6 | ζ3 | ζ65 | ζ32 | 1 | ζ6 | ζ65 | 1 | -1 | -1 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | complex lifted from C3×D4 |
| ρ15 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | complex lifted from C3×D4 |
| ρ16 | 6 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ17 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C2×F7 |
| ρ18 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | √7 | -√7 | orthogonal faithful |
| ρ19 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -√7 | √7 | orthogonal faithful |
►On 28 points - transitive group 28T23
Generators in S28
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)
(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)
(1 15)(2 18 3 21 5 20)(4 17 7 19 6 16)(8 22)(9 25 10 28 12 27)(11 24 14 26 13 23)
G:=sub<Sym(28)| (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (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), (1,15)(2,18,3,21,5,20)(4,17,7,19,6,16)(8,22)(9,25,10,28,12,27)(11,24,14,26,13,23)>;
G:=Group( (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (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), (1,15)(2,18,3,21,5,20)(4,17,7,19,6,16)(8,22)(9,25,10,28,12,27)(11,24,14,26,13,23) );
G=PermutationGroup([[(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21)], [(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)], [(1,15),(2,18,3,21,5,20),(4,17,7,19,6,16),(8,22),(9,25,10,28,12,27),(11,24,14,26,13,23)]])
G:=TransitiveGroup(28,23);
C4⋊F7 is a maximal subgroup of
C56⋊C6 D56⋊C3 D4⋊F7 Q8⋊2F7 D28⋊6C6 D4×F7 Q8⋊3F7
C4⋊F7 is a maximal quotient of C56⋊C6 D56⋊C3 C8.F7 C28⋊C12 D14⋊C12
Matrix representation of C4⋊F7 ►in GL6(𝔽3)
| 2 | 0 | 2 | 1 | 1 | 0 |
| 1 | 0 | 2 | 2 | 1 | 2 |
| 1 | 0 | 1 | 0 | 1 | 0 |
| 2 | 0 | 1 | 2 | 1 | 0 |
| 0 | 0 | 2 | 2 | 1 | 0 |
| 2 | 1 | 2 | 1 | 0 | 0 |
| 0 | 2 | 0 | 2 | 2 | 2 |
| 0 | 1 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 2 | 0 | 0 | 1 | 0 |
| 1 | 0 | 2 | 2 | 1 | 0 |
| 2 | 2 | 0 | 0 | 2 | 0 |
| 1 | 1 | 0 | 2 | 0 | 0 |
| 0 | 2 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 2 | 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(3))| [2,1,1,2,0,2,0,0,0,0,0,1,2,2,1,1,2,2,1,2,0,2,2,1,1,1,1,1,1,0,0,2,0,0,0,0],[0,0,1,1,1,2,2,1,1,2,0,2,0,0,0,0,2,0,2,1,1,0,2,0,2,1,0,1,1,2,2,0,0,0,0,0],[1,0,0,0,0,0,1,2,0,2,2,1,0,1,0,0,0,0,2,1,1,0,2,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C4⋊F7 in GAP, Magma, Sage, TeX
C_4\rtimes F_7
% in TeX
G:=Group("C4:F7"); // GroupNames label
G:=SmallGroup(168,9);
// by ID
G=gap.SmallGroup(168,9);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,141,66,3604,614]);
// Polycyclic
G:=Group<a,b,c|a^4=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C4⋊F7 in TeX
Character table of C4⋊F7 in TeX