Aliases: Q8⋊F7, D14.A4, D7⋊SL2(𝔽3), (Q8×D7)⋊2C3, (C7×Q8)⋊2C6, C14.A4⋊2C2, C14.2(C2×A4), C7⋊(C2×SL2(𝔽3)), C2.3(D7⋊A4), SmallGroup(336,135)
Series: Derived ►Chief ►Lower central ►Upper central
| C7×Q8 — Q8⋊F7 |
Generators and relations for Q8⋊F7
G = < a,b,c,d | a4=c7=d6=1, b2=a2, bab-1=a-1, ac=ca, dad-1=b, bc=cb, dbd-1=ab, dcd-1=c5 >
Character table of Q8⋊F7
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 7 | 14 | 28A | 28B | 28C | |
| size | 1 | 1 | 7 | 7 | 28 | 28 | 6 | 42 | 28 | 28 | 28 | 28 | 28 | 28 | 6 | 6 | 12 | 12 | 12 | |
| 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 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | -1 | ζ6 | ζ3 | ζ65 | ζ65 | ζ6 | ζ32 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | -1 | ζ65 | ζ32 | ζ6 | ζ6 | ζ65 | ζ3 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | -2 | 2 | -2 | -1 | -1 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | 2 | -2 | 0 | 0 | 0 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ8 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | 2 | -2 | 0 | 0 | 0 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | -2 | 2 | ζ65 | ζ6 | 0 | 0 | ζ3 | ζ32 | ζ6 | ζ32 | ζ65 | ζ3 | 2 | -2 | 0 | 0 | 0 | complex lifted from SL2(𝔽3) |
| ρ10 | 2 | -2 | -2 | 2 | ζ6 | ζ65 | 0 | 0 | ζ32 | ζ3 | ζ65 | ζ3 | ζ6 | ζ32 | 2 | -2 | 0 | 0 | 0 | complex lifted from SL2(𝔽3) |
| ρ11 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | 0 | 0 | ζ6 | ζ3 | ζ3 | ζ65 | ζ32 | ζ32 | 2 | -2 | 0 | 0 | 0 | complex lifted from SL2(𝔽3) |
| ρ12 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | 0 | 0 | ζ65 | ζ32 | ζ32 | ζ6 | ζ3 | ζ3 | 2 | -2 | 0 | 0 | 0 | complex lifted from SL2(𝔽3) |
| ρ13 | 3 | 3 | -3 | -3 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ15 | 6 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ16 | 6 | 6 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2ζ75+2ζ72+1 | 2ζ74+2ζ73+1 | 2ζ76+2ζ7+1 | orthogonal lifted from D7⋊A4 |
| ρ17 | 6 | 6 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2ζ76+2ζ7+1 | 2ζ75+2ζ72+1 | 2ζ74+2ζ73+1 | orthogonal lifted from D7⋊A4 |
| ρ18 | 6 | 6 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2ζ74+2ζ73+1 | 2ζ76+2ζ7+1 | 2ζ75+2ζ72+1 | orthogonal lifted from D7⋊A4 |
| ρ19 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 56 points
Generators in S56
(1 25 17 31)(2 26 18 32)(3 27 19 33)(4 28 20 34)(5 22 21 35)(6 23 15 29)(7 24 16 30)(8 43 51 37)(9 44 52 38)(10 45 53 39)(11 46 54 40)(12 47 55 41)(13 48 56 42)(14 49 50 36)
(1 39 17 45)(2 40 18 46)(3 41 19 47)(4 42 20 48)(5 36 21 49)(6 37 15 43)(7 38 16 44)(8 23 51 29)(9 24 52 30)(10 25 53 31)(11 26 54 32)(12 27 55 33)(13 28 56 34)(14 22 50 35)
(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 4 3 7 5 6)(8 46 34 12 44 35)(9 49 29 11 48 33)(10 45 31)(13 47 30 14 43 32)(15 18 20 19 16 21)(22 51 40 28 55 38)(23 54 42 27 52 36)(24 50 37 26 56 41)(25 53 39)
G:=sub<Sym(56)| (1,25,17,31)(2,26,18,32)(3,27,19,33)(4,28,20,34)(5,22,21,35)(6,23,15,29)(7,24,16,30)(8,43,51,37)(9,44,52,38)(10,45,53,39)(11,46,54,40)(12,47,55,41)(13,48,56,42)(14,49,50,36), (1,39,17,45)(2,40,18,46)(3,41,19,47)(4,42,20,48)(5,36,21,49)(6,37,15,43)(7,38,16,44)(8,23,51,29)(9,24,52,30)(10,25,53,31)(11,26,54,32)(12,27,55,33)(13,28,56,34)(14,22,50,35), (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), (2,4,3,7,5,6)(8,46,34,12,44,35)(9,49,29,11,48,33)(10,45,31)(13,47,30,14,43,32)(15,18,20,19,16,21)(22,51,40,28,55,38)(23,54,42,27,52,36)(24,50,37,26,56,41)(25,53,39)>;
G:=Group( (1,25,17,31)(2,26,18,32)(3,27,19,33)(4,28,20,34)(5,22,21,35)(6,23,15,29)(7,24,16,30)(8,43,51,37)(9,44,52,38)(10,45,53,39)(11,46,54,40)(12,47,55,41)(13,48,56,42)(14,49,50,36), (1,39,17,45)(2,40,18,46)(3,41,19,47)(4,42,20,48)(5,36,21,49)(6,37,15,43)(7,38,16,44)(8,23,51,29)(9,24,52,30)(10,25,53,31)(11,26,54,32)(12,27,55,33)(13,28,56,34)(14,22,50,35), (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), (2,4,3,7,5,6)(8,46,34,12,44,35)(9,49,29,11,48,33)(10,45,31)(13,47,30,14,43,32)(15,18,20,19,16,21)(22,51,40,28,55,38)(23,54,42,27,52,36)(24,50,37,26,56,41)(25,53,39) );
G=PermutationGroup([[(1,25,17,31),(2,26,18,32),(3,27,19,33),(4,28,20,34),(5,22,21,35),(6,23,15,29),(7,24,16,30),(8,43,51,37),(9,44,52,38),(10,45,53,39),(11,46,54,40),(12,47,55,41),(13,48,56,42),(14,49,50,36)], [(1,39,17,45),(2,40,18,46),(3,41,19,47),(4,42,20,48),(5,36,21,49),(6,37,15,43),(7,38,16,44),(8,23,51,29),(9,24,52,30),(10,25,53,31),(11,26,54,32),(12,27,55,33),(13,28,56,34),(14,22,50,35)], [(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)], [(2,4,3,7,5,6),(8,46,34,12,44,35),(9,49,29,11,48,33),(10,45,31),(13,47,30,14,43,32),(15,18,20,19,16,21),(22,51,40,28,55,38),(23,54,42,27,52,36),(24,50,37,26,56,41),(25,53,39)]])
Matrix representation of Q8⋊F7 ►in GL8(𝔽337)
| 0 | 336 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 250 | 0 | 195 | 219 | 219 | 195 |
| 0 | 0 | 142 | 55 | 142 | 0 | 24 | 24 |
| 0 | 0 | 313 | 118 | 31 | 118 | 313 | 0 |
| 0 | 0 | 0 | 313 | 118 | 31 | 118 | 313 |
| 0 | 0 | 24 | 24 | 0 | 142 | 55 | 142 |
| 0 | 0 | 195 | 219 | 219 | 195 | 0 | 250 |
| 208 | 209 | 0 | 0 | 0 | 0 | 0 | 0 |
| 209 | 129 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 118 | 313 | 313 | 118 |
| 0 | 0 | 219 | 250 | 219 | 0 | 195 | 195 |
| 0 | 0 | 142 | 24 | 55 | 24 | 142 | 0 |
| 0 | 0 | 0 | 142 | 24 | 55 | 24 | 142 |
| 0 | 0 | 195 | 195 | 0 | 219 | 250 | 219 |
| 0 | 0 | 118 | 313 | 313 | 118 | 0 | 31 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 336 | 336 | 336 | 336 | 336 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 336 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 208 | 209 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 336 | 336 | 336 | 336 | 336 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(337))| [0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,250,142,313,0,24,195,0,0,0,55,118,313,24,219,0,0,195,142,31,118,0,219,0,0,219,0,118,31,142,195,0,0,219,24,313,118,55,0,0,0,195,24,0,313,142,250],[208,209,0,0,0,0,0,0,209,129,0,0,0,0,0,0,0,0,31,219,142,0,195,118,0,0,0,250,24,142,195,313,0,0,118,219,55,24,0,313,0,0,313,0,24,55,219,118,0,0,313,195,142,24,250,0,0,0,118,195,0,142,219,31],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,1,0,0,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0],[336,208,0,0,0,0,0,0,0,209,0,0,0,0,0,0,0,0,1,0,0,0,336,0,0,0,0,0,0,1,336,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,0,336,1,0,0,0,1,0,0,336,0] >;
Q8⋊F7 in GAP, Magma, Sage, TeX
Q_8\rtimes F_7
% in TeX
G:=Group("Q8:F7"); // GroupNames label
G:=SmallGroup(336,135);
// by ID
G=gap.SmallGroup(336,135);
# by ID
G:=PCGroup([6,-2,-3,-2,2,-7,-2,116,518,225,735,357,4324,1450]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^7=d^6=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=b,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;
// generators/relations
Export
Subgroup lattice of Q8⋊F7 in TeX
Character table of Q8⋊F7 in TeX