metacyclic, supersoluble, monomial
Aliases: C4.F7, Dic14⋊C3, C28.1C6, Dic7.C6, C7⋊C3⋊Q8, C7⋊(C3×Q8), C7⋊C12.C2, C2.3(C2×F7), C14.1(C2×C6), (C4×C7⋊C3).1C2, (C2×C7⋊C3).1C22, SmallGroup(168,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.F7
G = < a,b,c | a4=b7=1, c6=a2, ab=ba, cac-1=a-1, cbc-1=b5 >
Character table of C4.F7
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 7 | 12A | 12B | 12C | 12D | 12E | 12F | 14 | 28A | 28B | |
| size | 1 | 1 | 7 | 7 | 2 | 14 | 14 | 7 | 7 | 6 | 14 | 14 | 14 | 14 | 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 | ζ32 | ζ3 | -1 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | 1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | 1 | -1 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | 1 | -1 | -1 | linear of order 6 |
| ρ10 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | 1 | -1 | -1 | linear of order 6 |
| ρ11 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | 1 | 1 | 1 | linear of order 6 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | 1 | 1 | 1 | linear of order 6 |
| ρ13 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | complex lifted from C3×Q8 |
| ρ15 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | complex lifted from C3×Q8 |
| ρ16 | 6 | 6 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C2×F7 |
| ρ17 | 6 | 6 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ18 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -√7 | √7 | symplectic faithful, Schur index 2 |
| ρ19 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | √7 | -√7 | symplectic faithful, Schur index 2 |
►On 56 points
Generators in S56
(1 8 3 6)(2 7 4 5)(9 50 15 56)(10 45 16 51)(11 52 17 46)(12 47 18 53)(13 54 19 48)(14 49 20 55)(21 40 27 34)(22 35 28 41)(23 42 29 36)(24 37 30 43)(25 44 31 38)(26 39 32 33)
(1 30 26 45 22 49 53)(2 50 46 31 54 23 27)(3 24 32 51 28 55 47)(4 56 52 25 48 29 21)(5 9 17 44 13 36 40)(6 37 33 10 41 14 18)(7 15 11 38 19 42 34)(8 43 39 16 35 20 12)
(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)| (1,8,3,6)(2,7,4,5)(9,50,15,56)(10,45,16,51)(11,52,17,46)(12,47,18,53)(13,54,19,48)(14,49,20,55)(21,40,27,34)(22,35,28,41)(23,42,29,36)(24,37,30,43)(25,44,31,38)(26,39,32,33), (1,30,26,45,22,49,53)(2,50,46,31,54,23,27)(3,24,32,51,28,55,47)(4,56,52,25,48,29,21)(5,9,17,44,13,36,40)(6,37,33,10,41,14,18)(7,15,11,38,19,42,34)(8,43,39,16,35,20,12), (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( (1,8,3,6)(2,7,4,5)(9,50,15,56)(10,45,16,51)(11,52,17,46)(12,47,18,53)(13,54,19,48)(14,49,20,55)(21,40,27,34)(22,35,28,41)(23,42,29,36)(24,37,30,43)(25,44,31,38)(26,39,32,33), (1,30,26,45,22,49,53)(2,50,46,31,54,23,27)(3,24,32,51,28,55,47)(4,56,52,25,48,29,21)(5,9,17,44,13,36,40)(6,37,33,10,41,14,18)(7,15,11,38,19,42,34)(8,43,39,16,35,20,12), (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([[(1,8,3,6),(2,7,4,5),(9,50,15,56),(10,45,16,51),(11,52,17,46),(12,47,18,53),(13,54,19,48),(14,49,20,55),(21,40,27,34),(22,35,28,41),(23,42,29,36),(24,37,30,43),(25,44,31,38),(26,39,32,33)], [(1,30,26,45,22,49,53),(2,50,46,31,54,23,27),(3,24,32,51,28,55,47),(4,56,52,25,48,29,21),(5,9,17,44,13,36,40),(6,37,33,10,41,14,18),(7,15,11,38,19,42,34),(8,43,39,16,35,20,12)], [(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)]])
C4.F7 is a maximal subgroup of
C56⋊C6 C8.F7 D4.F7 Q8.2F7 D28⋊6C6 D4⋊2F7 Q8×F7
C4.F7 is a maximal quotient of Dic7⋊C12 C28⋊C12
Matrix representation of C4.F7 ►in GL6(𝔽3)
| 2 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 2 | 0 | 0 | 2 |
| 2 | 2 | 2 | 0 | 0 | 1 |
| 2 | 0 | 0 | 1 | 2 | 2 |
| 2 | 0 | 0 | 2 | 2 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 2 | 2 | 1 | 2 | 1 |
| 0 | 2 | 2 | 2 | 1 | 0 |
| 0 | 2 | 2 | 2 | 0 | 1 |
| 2 | 1 | 1 | 2 | 0 | 1 |
| 0 | 1 | 0 | 1 | 2 | 0 |
| 0 | 2 | 0 | 1 | 2 | 0 |
| 0 | 1 | 0 | 1 | 0 | 1 |
| 1 | 2 | 0 | 1 | 2 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 2 | 2 | 2 |
G:=sub<GL(6,GF(3))| [2,0,2,2,2,1,0,1,2,0,0,0,0,2,2,0,0,0,0,0,0,1,2,0,0,0,0,2,2,0,1,2,1,2,0,1],[0,0,0,0,0,2,2,0,2,2,2,1,0,0,2,2,2,1,0,2,1,2,2,2,1,0,2,1,0,0,0,0,1,0,1,1],[0,0,0,1,1,0,1,2,1,2,0,0,0,0,0,0,0,1,1,1,1,1,0,2,2,2,0,2,1,2,0,0,1,1,1,2] >;
C4.F7 in GAP, Magma, Sage, TeX
C_4.F_7
% in TeX
G:=Group("C4.F7"); // GroupNames label
G:=SmallGroup(168,7);
// by ID
G=gap.SmallGroup(168,7);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,60,141,66,3604,614]);
// Polycyclic
G:=Group<a,b,c|a^4=b^7=1,c^6=a^2,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