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## G = D10.2F5order 400 = 24·52

### 2nd non-split extension by D10 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D10.2F5
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C52⋊4C8 — D10.2F5
 Lower central C52 — D10.2F5
 Upper central C1 — C2

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 >

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

Smallest permutation representation of D10.2F5
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`

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