Aliases: Q8.F7, Dic7.A4, C7⋊(C4.A4), C14.A4⋊1C2, C14.1(C2×A4), Q8⋊2D7⋊2C3, (C7×Q8).2C6, C2.2(D7⋊A4), SmallGroup(336,134)
Series: Derived ►Chief ►Lower central ►Upper central
| C7×Q8 — Q8.F7 |
Generators and relations for Q8.F7
G = < a,b,c,d | a4=c7=1, b2=d6=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 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 7 | 12A | 12B | 12C | 12D | 14 | 28A | 28B | 28C | |
| size | 1 | 1 | 42 | 28 | 28 | 6 | 7 | 7 | 28 | 28 | 6 | 28 | 28 | 28 | 28 | 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 | ζ3 | ζ32 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | -2 | 0 | -1 | -1 | 0 | 2i | -2i | 1 | 1 | 2 | i | -i | i | -i | -2 | 0 | 0 | 0 | complex lifted from C4.A4 |
| ρ8 | 2 | -2 | 0 | -1 | -1 | 0 | -2i | 2i | 1 | 1 | 2 | -i | i | -i | i | -2 | 0 | 0 | 0 | complex lifted from C4.A4 |
| ρ9 | 2 | -2 | 0 | ζ65 | ζ6 | 0 | -2i | 2i | ζ3 | ζ32 | 2 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | -2 | 0 | 0 | 0 | complex lifted from C4.A4 |
| ρ10 | 2 | -2 | 0 | ζ6 | ζ65 | 0 | -2i | 2i | ζ32 | ζ3 | 2 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | -2 | 0 | 0 | 0 | complex lifted from C4.A4 |
| ρ11 | 2 | -2 | 0 | ζ65 | ζ6 | 0 | 2i | -2i | ζ3 | ζ32 | 2 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | -2 | 0 | 0 | 0 | complex lifted from C4.A4 |
| ρ12 | 2 | -2 | 0 | ζ6 | ζ65 | 0 | 2i | -2i | ζ32 | ζ3 | 2 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | -2 | 0 | 0 | 0 | complex lifted from C4.A4 |
| ρ13 | 3 | 3 | -1 | 0 | 0 | -1 | 3 | 3 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ14 | 3 | 3 | 1 | 0 | 0 | -1 | -3 | -3 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | -1 | orthogonal lifted from C2×A4 |
| ρ15 | 6 | 6 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ16 | 6 | 6 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 2ζ75+2ζ72+1 | 2ζ76+2ζ7+1 | 2ζ74+2ζ73+1 | orthogonal lifted from D7⋊A4 |
| ρ17 | 6 | 6 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 2ζ76+2ζ7+1 | 2ζ74+2ζ73+1 | 2ζ75+2ζ72+1 | orthogonal lifted from D7⋊A4 |
| ρ18 | 6 | 6 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 2ζ74+2ζ73+1 | 2ζ75+2ζ72+1 | 2ζ76+2ζ7+1 | orthogonal lifted from D7⋊A4 |
| ρ19 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 112 points
Generators in S112
(1 12 3 6)(2 9 4 15)(5 7 11 13)(8 10 14 16)(17 108 23 102)(18 65 24 71)(19 57 25 63)(20 111 26 105)(21 68 27 74)(22 60 28 54)(29 51 35 45)(30 96 36 90)(31 77 37 83)(32 42 38 48)(33 99 39 93)(34 80 40 86)(41 91 47 97)(43 79 49 85)(44 94 50 100)(46 82 52 88)(53 112 59 106)(55 76 61 70)(56 103 62 109)(58 67 64 73)(66 110 72 104)(69 101 75 107)(78 98 84 92)(81 89 87 95)
(1 8 3 14)(2 5 4 11)(6 10 12 16)(7 9 13 15)(17 76 23 70)(18 56 24 62)(19 110 25 104)(20 67 26 73)(21 59 27 53)(22 101 28 107)(29 95 35 89)(30 88 36 82)(31 41 37 47)(32 98 38 92)(33 79 39 85)(34 44 40 50)(42 78 48 84)(43 93 49 99)(45 81 51 87)(46 96 52 90)(54 75 60 69)(55 102 61 108)(57 66 63 72)(58 105 64 111)(65 109 71 103)(68 112 74 106)(77 97 83 91)(80 100 86 94)
(1 81 77 106 85 110 102)(2 111 107 82 103 86 78)(3 87 83 112 79 104 108)(4 105 101 88 109 80 84)(5 58 22 30 65 94 48)(6 95 31 59 49 66 23)(7 67 60 96 24 50 32)(8 51 97 68 33 25 61)(9 26 69 52 62 34 98)(10 35 41 27 99 63 70)(11 64 28 36 71 100 42)(12 89 37 53 43 72 17)(13 73 54 90 18 44 38)(14 45 91 74 39 19 55)(15 20 75 46 56 40 92)(16 29 47 21 93 57 76)
(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 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76)(77 78 79 80 81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112)
G:=sub<Sym(112)| (1,12,3,6)(2,9,4,15)(5,7,11,13)(8,10,14,16)(17,108,23,102)(18,65,24,71)(19,57,25,63)(20,111,26,105)(21,68,27,74)(22,60,28,54)(29,51,35,45)(30,96,36,90)(31,77,37,83)(32,42,38,48)(33,99,39,93)(34,80,40,86)(41,91,47,97)(43,79,49,85)(44,94,50,100)(46,82,52,88)(53,112,59,106)(55,76,61,70)(56,103,62,109)(58,67,64,73)(66,110,72,104)(69,101,75,107)(78,98,84,92)(81,89,87,95), (1,8,3,14)(2,5,4,11)(6,10,12,16)(7,9,13,15)(17,76,23,70)(18,56,24,62)(19,110,25,104)(20,67,26,73)(21,59,27,53)(22,101,28,107)(29,95,35,89)(30,88,36,82)(31,41,37,47)(32,98,38,92)(33,79,39,85)(34,44,40,50)(42,78,48,84)(43,93,49,99)(45,81,51,87)(46,96,52,90)(54,75,60,69)(55,102,61,108)(57,66,63,72)(58,105,64,111)(65,109,71,103)(68,112,74,106)(77,97,83,91)(80,100,86,94), (1,81,77,106,85,110,102)(2,111,107,82,103,86,78)(3,87,83,112,79,104,108)(4,105,101,88,109,80,84)(5,58,22,30,65,94,48)(6,95,31,59,49,66,23)(7,67,60,96,24,50,32)(8,51,97,68,33,25,61)(9,26,69,52,62,34,98)(10,35,41,27,99,63,70)(11,64,28,36,71,100,42)(12,89,37,53,43,72,17)(13,73,54,90,18,44,38)(14,45,91,74,39,19,55)(15,20,75,46,56,40,92)(16,29,47,21,93,57,76), (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)(53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112)>;
G:=Group( (1,12,3,6)(2,9,4,15)(5,7,11,13)(8,10,14,16)(17,108,23,102)(18,65,24,71)(19,57,25,63)(20,111,26,105)(21,68,27,74)(22,60,28,54)(29,51,35,45)(30,96,36,90)(31,77,37,83)(32,42,38,48)(33,99,39,93)(34,80,40,86)(41,91,47,97)(43,79,49,85)(44,94,50,100)(46,82,52,88)(53,112,59,106)(55,76,61,70)(56,103,62,109)(58,67,64,73)(66,110,72,104)(69,101,75,107)(78,98,84,92)(81,89,87,95), (1,8,3,14)(2,5,4,11)(6,10,12,16)(7,9,13,15)(17,76,23,70)(18,56,24,62)(19,110,25,104)(20,67,26,73)(21,59,27,53)(22,101,28,107)(29,95,35,89)(30,88,36,82)(31,41,37,47)(32,98,38,92)(33,79,39,85)(34,44,40,50)(42,78,48,84)(43,93,49,99)(45,81,51,87)(46,96,52,90)(54,75,60,69)(55,102,61,108)(57,66,63,72)(58,105,64,111)(65,109,71,103)(68,112,74,106)(77,97,83,91)(80,100,86,94), (1,81,77,106,85,110,102)(2,111,107,82,103,86,78)(3,87,83,112,79,104,108)(4,105,101,88,109,80,84)(5,58,22,30,65,94,48)(6,95,31,59,49,66,23)(7,67,60,96,24,50,32)(8,51,97,68,33,25,61)(9,26,69,52,62,34,98)(10,35,41,27,99,63,70)(11,64,28,36,71,100,42)(12,89,37,53,43,72,17)(13,73,54,90,18,44,38)(14,45,91,74,39,19,55)(15,20,75,46,56,40,92)(16,29,47,21,93,57,76), (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)(53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112) );
G=PermutationGroup([[(1,12,3,6),(2,9,4,15),(5,7,11,13),(8,10,14,16),(17,108,23,102),(18,65,24,71),(19,57,25,63),(20,111,26,105),(21,68,27,74),(22,60,28,54),(29,51,35,45),(30,96,36,90),(31,77,37,83),(32,42,38,48),(33,99,39,93),(34,80,40,86),(41,91,47,97),(43,79,49,85),(44,94,50,100),(46,82,52,88),(53,112,59,106),(55,76,61,70),(56,103,62,109),(58,67,64,73),(66,110,72,104),(69,101,75,107),(78,98,84,92),(81,89,87,95)], [(1,8,3,14),(2,5,4,11),(6,10,12,16),(7,9,13,15),(17,76,23,70),(18,56,24,62),(19,110,25,104),(20,67,26,73),(21,59,27,53),(22,101,28,107),(29,95,35,89),(30,88,36,82),(31,41,37,47),(32,98,38,92),(33,79,39,85),(34,44,40,50),(42,78,48,84),(43,93,49,99),(45,81,51,87),(46,96,52,90),(54,75,60,69),(55,102,61,108),(57,66,63,72),(58,105,64,111),(65,109,71,103),(68,112,74,106),(77,97,83,91),(80,100,86,94)], [(1,81,77,106,85,110,102),(2,111,107,82,103,86,78),(3,87,83,112,79,104,108),(4,105,101,88,109,80,84),(5,58,22,30,65,94,48),(6,95,31,59,49,66,23),(7,67,60,96,24,50,32),(8,51,97,68,33,25,61),(9,26,69,52,62,34,98),(10,35,41,27,99,63,70),(11,64,28,36,71,100,42),(12,89,37,53,43,72,17),(13,73,54,90,18,44,38),(14,45,91,74,39,19,55),(15,20,75,46,56,40,92),(16,29,47,21,93,57,76)], [(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),(53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76),(77,78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112)]])
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 | 55 | 0 | 24 | 142 | 142 | 24 |
| 0 | 0 | 313 | 31 | 313 | 0 | 118 | 118 |
| 0 | 0 | 219 | 195 | 250 | 195 | 219 | 0 |
| 0 | 0 | 0 | 219 | 195 | 250 | 195 | 219 |
| 0 | 0 | 118 | 118 | 0 | 313 | 31 | 313 |
| 0 | 0 | 24 | 142 | 142 | 24 | 0 | 55 |
| 128 | 129 | 0 | 0 | 0 | 0 | 0 | 0 |
| 129 | 209 | 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 |
| 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 |
| 189 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 220 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 17 | 17 | 0 | 17 | 0 |
| 0 | 0 | 0 | 320 | 0 | 320 | 320 | 14 |
| 0 | 0 | 17 | 0 | 0 | 31 | 17 | 17 |
| 0 | 0 | 0 | 31 | 17 | 17 | 0 | 17 |
| 0 | 0 | 323 | 323 | 306 | 323 | 306 | 306 |
| 0 | 0 | 320 | 0 | 320 | 320 | 14 | 0 |
G:=sub<GL(8,GF(337))| [0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,55,313,219,0,118,24,0,0,0,31,195,219,118,142,0,0,24,313,250,195,0,142,0,0,142,0,195,250,313,24,0,0,142,118,219,195,31,0,0,0,24,118,0,219,313,55],[128,129,0,0,0,0,0,0,129,209,0,0,0,0,0,0,0,0,250,142,313,0,24,195,0,0,0,55,118,313,24,219,0,0,195,142,31,118,0,219,0,0,219,0,118,31,142,195,0,0,219,24,313,118,55,0,0,0,195,24,0,313,142,250],[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],[189,72,0,0,0,0,0,0,0,220,0,0,0,0,0,0,0,0,31,0,17,0,323,320,0,0,17,320,0,31,323,0,0,0,17,0,0,17,306,320,0,0,0,320,31,17,323,320,0,0,17,320,17,0,306,14,0,0,0,14,17,17,306,0] >;
Q8.F7 in GAP, Magma, Sage, TeX
Q_8.F_7
% in TeX
G:=Group("Q8.F7"); // GroupNames label
G:=SmallGroup(336,134);
// by ID
G=gap.SmallGroup(336,134);
# by ID
G:=PCGroup([6,-2,-3,-2,2,-7,-2,1008,116,518,225,735,357,4324,1450]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^7=1,b^2=d^6=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