metabelian, supersoluble, monomial, A-group
Aliases: D10.2F5, Dic5.2F5, D5⋊(C5⋊C8), (C5×D5)⋊2C8, C52⋊4(C2×C8), C5⋊3(D5⋊C8), C10.4(C2×F5), C52⋊4C8⋊1C2, C52⋊5C8⋊1C2, (D5×C10).3C4, C2.2(D5⋊F5), (D5×Dic5).7C2, (C5×Dic5).4C4, C52⋊6C4.4C22, C5⋊1(C2×C5⋊C8), (C5×C10).11(C2×C4), SmallGroup(400,127)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — D10.2F5 |
Generators and relations for D10.2F5
G = < a,b,c,d | a10=b2=c5=1, d4=a5, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c3 >
5C2
5C2
2C5
2C5
5C4
5C22
25C4
2C10
2C10
5C10
5C10
25C2×C4
25C8
25C8
5C20
5Dic5
5Dic5
10Dic5
10Dic5
25C2×C8
10C5⋊C8
10C5⋊C8
Character table of D10.2F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | |
| size | 1 | 1 | 5 | 5 | 5 | 5 | 25 | 25 | 4 | 4 | 8 | 8 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 4 | 4 | 8 | 8 | 20 | 20 | 20 | 20 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | -i | i | i | i | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | i | -i | -i | -i | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | i | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | -1 | -1 | -1 | -1 | 1 | -1 | i | -i | linear of order 8 |
| ρ10 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | -1 | -1 | -1 | -1 | 1 | -1 | -i | i | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | ζ87 | ζ85 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | ζ8 | ζ83 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | linear of order 8 |
| ρ13 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | -1 | -1 | -1 | -1 | 1 | -1 | i | -i | linear of order 8 |
| ρ14 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | -1 | -1 | -1 | -1 | 1 | -1 | -i | i | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | ζ85 | ζ87 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | linear of order 8 |
| ρ16 | 1 | -1 | 1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | ζ83 | ζ8 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | linear of order 8 |
| ρ17 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ18 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from F5 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×F5 |
| ρ20 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -4 | 1 | 1 | 1 | -1 | 0 | 0 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -4 | 1 | 1 | -1 | 1 | 0 | 0 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 1 | 1 | 1 | 0 | 0 | i | -i | complex lifted from D5⋊C8, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 1 | 1 | 1 | 0 | 0 | -i | i | complex lifted from D5⋊C8, Schur index 2 |
| ρ25 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | orthogonal lifted from D5⋊F5 |
| ρ26 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5⋊F5 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -3 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -3 | 2 | 0 | 0 | 0 | 0 | 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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 30)(10 29)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 40)(20 39)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 70)(50 69)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 80)(60 79)
(1 7 3 9 5)(2 8 4 10 6)(11 17 13 19 15)(12 18 14 20 16)(21 25 29 23 27)(22 26 30 24 28)(31 35 39 33 37)(32 36 40 34 38)(41 47 43 49 45)(42 48 44 50 46)(51 57 53 59 55)(52 58 54 60 56)(61 65 69 63 67)(62 66 70 64 68)(71 75 79 73 77)(72 76 80 74 78)
(1 42 19 58 6 47 14 53)(2 49 18 51 7 44 13 56)(3 46 17 54 8 41 12 59)(4 43 16 57 9 48 11 52)(5 50 15 60 10 45 20 55)(21 62 39 78 26 67 34 73)(22 69 38 71 27 64 33 76)(23 66 37 74 28 61 32 79)(24 63 36 77 29 68 31 72)(25 70 35 80 30 65 40 75)
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,30)(10,29)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,40)(20,39)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,70)(50,69)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,80)(60,79), (1,7,3,9,5)(2,8,4,10,6)(11,17,13,19,15)(12,18,14,20,16)(21,25,29,23,27)(22,26,30,24,28)(31,35,39,33,37)(32,36,40,34,38)(41,47,43,49,45)(42,48,44,50,46)(51,57,53,59,55)(52,58,54,60,56)(61,65,69,63,67)(62,66,70,64,68)(71,75,79,73,77)(72,76,80,74,78), (1,42,19,58,6,47,14,53)(2,49,18,51,7,44,13,56)(3,46,17,54,8,41,12,59)(4,43,16,57,9,48,11,52)(5,50,15,60,10,45,20,55)(21,62,39,78,26,67,34,73)(22,69,38,71,27,64,33,76)(23,66,37,74,28,61,32,79)(24,63,36,77,29,68,31,72)(25,70,35,80,30,65,40,75)>;
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,30)(10,29)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,40)(20,39)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,70)(50,69)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,80)(60,79), (1,7,3,9,5)(2,8,4,10,6)(11,17,13,19,15)(12,18,14,20,16)(21,25,29,23,27)(22,26,30,24,28)(31,35,39,33,37)(32,36,40,34,38)(41,47,43,49,45)(42,48,44,50,46)(51,57,53,59,55)(52,58,54,60,56)(61,65,69,63,67)(62,66,70,64,68)(71,75,79,73,77)(72,76,80,74,78), (1,42,19,58,6,47,14,53)(2,49,18,51,7,44,13,56)(3,46,17,54,8,41,12,59)(4,43,16,57,9,48,11,52)(5,50,15,60,10,45,20,55)(21,62,39,78,26,67,34,73)(22,69,38,71,27,64,33,76)(23,66,37,74,28,61,32,79)(24,63,36,77,29,68,31,72)(25,70,35,80,30,65,40,75) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,30),(10,29),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,40),(20,39),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,70),(50,69),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,80),(60,79)], [(1,7,3,9,5),(2,8,4,10,6),(11,17,13,19,15),(12,18,14,20,16),(21,25,29,23,27),(22,26,30,24,28),(31,35,39,33,37),(32,36,40,34,38),(41,47,43,49,45),(42,48,44,50,46),(51,57,53,59,55),(52,58,54,60,56),(61,65,69,63,67),(62,66,70,64,68),(71,75,79,73,77),(72,76,80,74,78)], [(1,42,19,58,6,47,14,53),(2,49,18,51,7,44,13,56),(3,46,17,54,8,41,12,59),(4,43,16,57,9,48,11,52),(5,50,15,60,10,45,20,55),(21,62,39,78,26,67,34,73),(22,69,38,71,27,64,33,76),(23,66,37,74,28,61,32,79),(24,63,36,77,29,68,31,72),(25,70,35,80,30,65,40,75)]])
Matrix representation of D10.2F5 ►in GL8(𝔽41)
| 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 40 |
| 1 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 40 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 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 | 40 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
G:=sub<GL(8,GF(41))| [0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0] >;
D10.2F5 in GAP, Magma, Sage, TeX
D_{10}._2F_5 % in TeX
G:=Group("D10.2F5"); // GroupNames label
G:=SmallGroup(400,127);
// by ID
G=gap.SmallGroup(400,127);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,50,1444,970,496,8645,2897]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^5=1,d^4=a^5,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of D10.2F5 in TeX
Character table of D10.2F5 in TeX