metabelian, supersoluble, monomial
Aliases: Q8⋊2F7, D28.1C6, Q8⋊D7⋊C3, C7⋊C8⋊3C6, C7⋊C24⋊3C2, C7⋊C3⋊3SD16, (C7×Q8)⋊3C6, C7⋊3(C3×SD16), C4⋊F7.1C2, C4.3(C2×F7), C28.3(C2×C6), C14.9(C3×D4), C2.6(Dic7⋊C6), (Q8×C7⋊C3)⋊1C2, (C2×C7⋊C3).9D4, (C4×C7⋊C3).3C22, SmallGroup(336,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2F7
G = < a,b,c,d | a4=c7=d6=1, b2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c5 >
Character table of Q8⋊2F7
| 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 | 28C | |
| size | 1 | 1 | 28 | 7 | 7 | 2 | 4 | 7 | 7 | 28 | 28 | 6 | 14 | 14 | 14 | 14 | 28 | 28 | 6 | 14 | 14 | 14 | 14 | 12 | 12 | 12 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | 1 | -1 | linear of order 6 |
| ρ8 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | -1 | 1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ10 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | 1 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | 1 | -1 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | -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 | 0 | -2 | 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 | 0 | -2 | 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 | 0 | -2 | 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 | 0 | 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 | 0 | 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 | ζ87ζ32+ζ85ζ32 | ζ87ζ3+ζ85ζ3 | ζ83ζ32+ζ8ζ32 | ζ83ζ3+ζ8ζ3 | 0 | 0 | 0 | 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 | ζ83ζ32+ζ8ζ32 | ζ87ζ3+ζ85ζ3 | ζ87ζ32+ζ85ζ32 | 0 | 0 | 0 | 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 | ζ87ζ3+ζ85ζ3 | ζ87ζ32+ζ85ζ32 | ζ83ζ3+ζ8ζ3 | ζ83ζ32+ζ8ζ32 | 0 | 0 | 0 | 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 | ζ83ζ32+ζ8ζ32 | ζ83ζ3+ζ8ζ3 | ζ87ζ32+ζ85ζ32 | ζ87ζ3+ζ85ζ3 | 0 | 0 | 0 | complex lifted from C3×SD16 |
| ρ22 | 6 | 6 | 0 | 0 | 0 | 6 | -6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | orthogonal lifted from C2×F7 |
| ρ23 | 6 | 6 | 0 | 0 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from 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 | -√-7 | 1 | √-7 | complex lifted from Dic7⋊C6 |
| ρ25 | 6 | 6 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | √-7 | 1 | -√-7 | complex lifted from Dic7⋊C6 |
| ρ26 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 56 points
Generators in S56
(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)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)
(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)(9 11 10 14 12 13)(15 22)(16 25 17 28 19 27)(18 24 21 26 20 23)(29 43)(30 46 31 49 33 48)(32 45 35 47 34 44)(36 50)(37 53 38 56 40 55)(39 52 42 54 41 51)
G:=sub<Sym(56)| (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)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49), (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)(9,11,10,14,12,13)(15,22)(16,25,17,28,19,27)(18,24,21,26,20,23)(29,43)(30,46,31,49,33,48)(32,45,35,47,34,44)(36,50)(37,53,38,56,40,55)(39,52,42,54,41,51)>;
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)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49), (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)(9,11,10,14,12,13)(15,22)(16,25,17,28,19,27)(18,24,21,26,20,23)(29,43)(30,46,31,49,33,48)(32,45,35,47,34,44)(36,50)(37,53,38,56,40,55)(39,52,42,54,41,51) );
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),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49)], [(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),(9,11,10,14,12,13),(15,22),(16,25,17,28,19,27),(18,24,21,26,20,23),(29,43),(30,46,31,49,33,48),(32,45,35,47,34,44),(36,50),(37,53,38,56,40,55),(39,52,42,54,41,51)]])
Matrix representation of Q8⋊2F7 ►in GL8(𝔽337)
| 1 | 336 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 336 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 98 | 0 | 0 | 0 | 0 | 0 | 0 |
| 196 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 336 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 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 | 1 | 0 | 336 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 336 |
| 128 | 209 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 209 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,GF(337))| [1,2,0,0,0,0,0,0,336,336,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,196,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,336,336,336,336,336,336],[128,0,0,0,0,0,0,0,209,209,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
Q8⋊2F7 in GAP, Magma, Sage, TeX
Q_8\rtimes_2F_7
% in TeX
G:=Group("Q8:2F7"); // GroupNames label
G:=SmallGroup(336,20);
// by ID
G=gap.SmallGroup(336,20);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,151,867,441,69,10373,1745]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^7=d^6=1,b^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^5>;
// generators/relations
Export
Subgroup lattice of Q8⋊2F7 in TeX
Character table of Q8⋊2F7 in TeX