metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊4F5, Dic20⋊2C4, C5⋊C8.1D4, (C5×D8)⋊3C4, C8.6(C2×F5), C40.5(C2×C4), D4.D5⋊2C4, C8⋊F5⋊2C2, D4.F5⋊2C2, D4.2(C2×F5), C2.18(D4×F5), C5⋊1(C8.26D4), D4⋊F5⋊2C2, C10.17(C4×D4), C40.C4⋊2C2, C4.4(C22×F5), D8⋊3D5.3C2, C20.4(C22×C4), D5⋊C8.2C22, (C4×F5).2C22, D10.2(C4○D4), C4.F5.2C22, Dic10.2(C2×C4), Dic5.73(C2×D4), (C4×D5).26C23, (C8×D5).14C22, D4⋊2D5.5C22, C5⋊2C8.8(C2×C4), (C5×D4).2(C2×C4), SmallGroup(320,1071)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a2b, dcd-1=c3 >
Subgroups: 386 in 104 conjugacy classes, 40 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C42, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C5⋊2C8, C40, C5⋊C8, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, C8.26D4, C8×D5, Dic20, D4.D5, C5×D8, D5⋊C8, C4.F5, C4×F5, C2×C5⋊C8, C22.F5, D4⋊2D5, C8⋊F5, C40.C4, D4⋊F5, D8⋊3D5, D4.F5, D8⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5, C8.26D4, C22×F5, D4×F5, D8⋊F5
Character table of D8⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 20 | 40A | 40B | |
| size | 1 | 1 | 4 | 4 | 10 | 2 | 5 | 5 | 20 | 20 | 20 | 20 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 4 | 16 | 16 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 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 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 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 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | i | -1 | -i | -1 | 1 | 1 | -i | i | i | -i | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | -1 | -i | 1 | 1 | -1 | i | -i | -i | i | 1 | -i | i | i | -i | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -i | -1 | i | -1 | 1 | 1 | i | -i | -i | i | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | -1 | i | 1 | 1 | -1 | -i | i | i | -i | 1 | i | -i | -i | i | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -i | 1 | i | -1 | 1 | -1 | -i | i | i | -i | 1 | -i | i | i | -i | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -i | 1 | i | 1 | 1 | 1 | i | -i | -i | i | -1 | -i | i | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | i | 1 | -i | -1 | 1 | -1 | i | -i | -i | i | 1 | i | -i | -i | i | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | i | 1 | -i | 1 | 1 | 1 | -i | i | i | -i | -1 | i | -i | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ17 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | -4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | 4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ24 | 4 | 4 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.26D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.26D4 |
| ρ27 | 8 | 8 | 0 | 0 | 0 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4×F5 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -√10 | √10 | symplectic faithful, Schur index 2 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | √10 | -√10 | symplectic faithful, Schur index 2 |
►On 80 points
Generators in S80
(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)
(1 59)(2 58)(3 57)(4 64)(5 63)(6 62)(7 61)(8 60)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 80)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)
(1 75 41 12 66)(2 76 42 13 67)(3 77 43 14 68)(4 78 44 15 69)(5 79 45 16 70)(6 80 46 9 71)(7 73 47 10 72)(8 74 48 11 65)(17 31 54 60 38)(18 32 55 61 39)(19 25 56 62 40)(20 26 49 63 33)(21 27 50 64 34)(22 28 51 57 35)(23 29 52 58 36)(24 30 53 59 37)
(2 6)(4 8)(9 76 46 67)(10 73 47 72)(11 78 48 69)(12 75 41 66)(13 80 42 71)(14 77 43 68)(15 74 44 65)(16 79 45 70)(17 52 27 40)(18 49 28 37)(19 54 29 34)(20 51 30 39)(21 56 31 36)(22 53 32 33)(23 50 25 38)(24 55 26 35)(57 59 61 63)(58 64 62 60)
G:=sub<Sym(80)| (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), (1,59)(2,58)(3,57)(4,64)(5,63)(6,62)(7,61)(8,60)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,80)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,72)(56,71), (1,75,41,12,66)(2,76,42,13,67)(3,77,43,14,68)(4,78,44,15,69)(5,79,45,16,70)(6,80,46,9,71)(7,73,47,10,72)(8,74,48,11,65)(17,31,54,60,38)(18,32,55,61,39)(19,25,56,62,40)(20,26,49,63,33)(21,27,50,64,34)(22,28,51,57,35)(23,29,52,58,36)(24,30,53,59,37), (2,6)(4,8)(9,76,46,67)(10,73,47,72)(11,78,48,69)(12,75,41,66)(13,80,42,71)(14,77,43,68)(15,74,44,65)(16,79,45,70)(17,52,27,40)(18,49,28,37)(19,54,29,34)(20,51,30,39)(21,56,31,36)(22,53,32,33)(23,50,25,38)(24,55,26,35)(57,59,61,63)(58,64,62,60)>;
G:=Group( (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), (1,59)(2,58)(3,57)(4,64)(5,63)(6,62)(7,61)(8,60)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,80)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,72)(56,71), (1,75,41,12,66)(2,76,42,13,67)(3,77,43,14,68)(4,78,44,15,69)(5,79,45,16,70)(6,80,46,9,71)(7,73,47,10,72)(8,74,48,11,65)(17,31,54,60,38)(18,32,55,61,39)(19,25,56,62,40)(20,26,49,63,33)(21,27,50,64,34)(22,28,51,57,35)(23,29,52,58,36)(24,30,53,59,37), (2,6)(4,8)(9,76,46,67)(10,73,47,72)(11,78,48,69)(12,75,41,66)(13,80,42,71)(14,77,43,68)(15,74,44,65)(16,79,45,70)(17,52,27,40)(18,49,28,37)(19,54,29,34)(20,51,30,39)(21,56,31,36)(22,53,32,33)(23,50,25,38)(24,55,26,35)(57,59,61,63)(58,64,62,60) );
G=PermutationGroup([[(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)], [(1,59),(2,58),(3,57),(4,64),(5,63),(6,62),(7,61),(8,60),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,42),(24,41),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,80),(49,70),(50,69),(51,68),(52,67),(53,66),(54,65),(55,72),(56,71)], [(1,75,41,12,66),(2,76,42,13,67),(3,77,43,14,68),(4,78,44,15,69),(5,79,45,16,70),(6,80,46,9,71),(7,73,47,10,72),(8,74,48,11,65),(17,31,54,60,38),(18,32,55,61,39),(19,25,56,62,40),(20,26,49,63,33),(21,27,50,64,34),(22,28,51,57,35),(23,29,52,58,36),(24,30,53,59,37)], [(2,6),(4,8),(9,76,46,67),(10,73,47,72),(11,78,48,69),(12,75,41,66),(13,80,42,71),(14,77,43,68),(15,74,44,65),(16,79,45,70),(17,52,27,40),(18,49,28,37),(19,54,29,34),(20,51,30,39),(21,56,31,36),(22,53,32,33),(23,50,25,38),(24,55,26,35),(57,59,61,63),(58,64,62,60)]])
Matrix representation of D8⋊F5 ►in GL8(𝔽41)
| 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 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 40 | 40 | 40 | 40 | 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 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40] >;
D8⋊F5 in GAP, Magma, Sage, TeX
D_8\rtimes F_5
% in TeX
G:=Group("D8:F5"); // GroupNames label
G:=SmallGroup(320,1071);
// by ID
G=gap.SmallGroup(320,1071);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,851,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations
Export