metabelian, supersoluble, monomial
Aliases: D4⋊F7, D28⋊1C6, D4⋊D7⋊C3, C7⋊C3⋊2D8, C7⋊C8⋊1C6, C7⋊2(C3×D8), C7⋊C24⋊1C2, C4⋊F7⋊1C2, (C7×D4)⋊1C6, C4.1(C2×F7), C28.1(C2×C6), C14.7(C3×D4), C2.4(Dic7⋊C6), (D4×C7⋊C3)⋊1C2, (C2×C7⋊C3).7D4, (C4×C7⋊C3).1C22, SmallGroup(336,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊F7
G = < a,b,c,d | a4=b2=c7=d6=1, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c5 >
Character table of D4⋊F7
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7 | 8A | 8B | 12A | 12B | 14A | 14B | 14C | 24A | 24B | 24C | 24D | 28 | |
| size | 1 | 1 | 4 | 28 | 7 | 7 | 2 | 7 | 7 | 28 | 28 | 28 | 28 | 6 | 14 | 14 | 14 | 14 | 6 | 12 | 12 | 14 | 14 | 14 | 14 | 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 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ3 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | -1 | -1 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ32 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | -1 | -1 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ65 | ζ3 | ζ32 | ζ6 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ6 | ζ32 | ζ3 | ζ65 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | linear of order 3 |
| ρ11 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | linear of order 6 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1+√-3 | 1-√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | complex lifted from C3×D4 |
| ρ17 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1-√-3 | 1+√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | complex lifted from C3×D4 |
| ρ18 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | 0 | 0 | -2 | 0 | 0 | ζ83ζ3-ζ8ζ3 | ζ87ζ3-ζ85ζ3 | ζ83ζ32-ζ8ζ32 | ζ87ζ32-ζ85ζ32 | 0 | complex lifted from C3×D8 |
| ρ19 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | 0 | 0 | -2 | 0 | 0 | ζ83ζ32-ζ8ζ32 | ζ87ζ32-ζ85ζ32 | ζ83ζ3-ζ8ζ3 | ζ87ζ3-ζ85ζ3 | 0 | complex lifted from C3×D8 |
| ρ20 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | 0 | 0 | -2 | 0 | 0 | ζ87ζ32-ζ85ζ32 | ζ83ζ32-ζ8ζ32 | ζ87ζ3-ζ85ζ3 | ζ83ζ3-ζ8ζ3 | 0 | complex lifted from C3×D8 |
| ρ21 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | 0 | 0 | -2 | 0 | 0 | ζ87ζ3-ζ85ζ3 | ζ83ζ3-ζ8ζ3 | ζ87ζ32-ζ85ζ32 | ζ83ζ32-ζ8ζ32 | 0 | complex lifted from C3×D8 |
| ρ22 | 6 | 6 | -6 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×F7 |
| ρ23 | 6 | 6 | 6 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from F7 |
| ρ24 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | √-7 | -√-7 | 0 | 0 | 0 | 0 | 1 | complex lifted from Dic7⋊C6 |
| ρ25 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -√-7 | √-7 | 0 | 0 | 0 | 0 | 1 | complex lifted from Dic7⋊C6 |
| ρ26 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►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 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(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)
(1 34)(2 30 3 33 5 32)(4 29 7 31 6 35)(8 41)(9 37 10 40 12 39)(11 36 14 38 13 42)(15 48)(16 44 17 47 19 46)(18 43 21 45 20 49)(22 55)(23 51 24 54 26 53)(25 50 28 52 27 56)
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), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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), (1,34)(2,30,3,33,5,32)(4,29,7,31,6,35)(8,41)(9,37,10,40,12,39)(11,36,14,38,13,42)(15,48)(16,44,17,47,19,46)(18,43,21,45,20,49)(22,55)(23,51,24,54,26,53)(25,50,28,52,27,56)>;
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), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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), (1,34)(2,30,3,33,5,32)(4,29,7,31,6,35)(8,41)(9,37,10,40,12,39)(11,36,14,38,13,42)(15,48)(16,44,17,47,19,46)(18,43,21,45,20,49)(22,55)(23,51,24,54,26,53)(25,50,28,52,27,56) );
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)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(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)], [(1,34),(2,30,3,33,5,32),(4,29,7,31,6,35),(8,41),(9,37,10,40,12,39),(11,36,14,38,13,42),(15,48),(16,44,17,47,19,46),(18,43,21,45,20,49),(22,55),(23,51,24,54,26,53),(25,50,28,52,27,56)]])
Matrix representation of D4⋊F7 ►in GL8(𝔽337)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 336 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 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 | 64 | 325 | 261 | 1 | 0 | 0 |
| 0 | 0 | 64 | 325 | 261 | 0 | 1 | 0 |
| 0 | 0 | 64 | 325 | 261 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 212 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 212 | 1 | 0 | 0 | 0 |
| 0 | 0 | 336 | 213 | 1 | 0 | 0 | 0 |
| 0 | 0 | 26 | 217 | 311 | 0 | 1 | 0 |
| 0 | 0 | 26 | 217 | 311 | 0 | 0 | 1 |
| 0 | 0 | 243 | 71 | 285 | 1 | 125 | 124 |
| 21 | 316 | 0 | 0 | 0 | 0 | 0 | 0 |
| 316 | 316 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 336 | 0 | 1 |
| 0 | 0 | 165 | 285 | 120 | 123 | 336 | 336 |
| 0 | 0 | 120 | 146 | 26 | 211 | 213 | 1 |
| 0 | 0 | 79 | 259 | 180 | 26 | 0 | 0 |
| 0 | 0 | 79 | 260 | 180 | 26 | 0 | 0 |
| 0 | 0 | 292 | 260 | 305 | 26 | 0 | 0 |
G:=sub<GL(8,GF(337))| [0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,336,0,0,64,64,64,0,0,0,336,0,325,325,325,0,0,0,0,336,261,261,261,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,0,336,26,26,243,0,0,212,212,213,217,217,71,0,0,1,1,1,311,311,285,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,125,0,0,0,0,0,0,1,124],[21,316,0,0,0,0,0,0,316,316,0,0,0,0,0,0,0,0,0,165,120,79,79,292,0,0,0,285,146,259,260,260,0,0,0,120,26,180,180,305,0,0,336,123,211,26,26,26,0,0,0,336,213,0,0,0,0,0,1,336,1,0,0,0] >;
D4⋊F7 in GAP, Magma, Sage, TeX
D_4\rtimes F_7
% in TeX
G:=Group("D4:F7"); // GroupNames label
G:=SmallGroup(336,18);
// by ID
G=gap.SmallGroup(336,18);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,867,441,69,10373,1745]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^7=d^6=1,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;
// generators/relations
Export
Subgroup lattice of D4⋊F7 in TeX
Character table of D4⋊F7 in TeX