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## G = D10.F5order 400 = 24·52

### 1st non-split extension by D10 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D10.F5
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — C5×C5⋊C8 — D10.F5
 Lower central C52 — C5×C10 — D10.F5
 Upper central C1 — C2

Generators and relations for D10.F5
G = < a,b,c,d | a10=b2=c5=1, d4=a5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c3 >

Smallest permutation representation of D10.F5
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 22)(2 21)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 40)(20 39)(41 63)(42 62)(43 61)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 73)(52 72)(53 71)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)
(1 3 5 7 9)(2 4 6 8 10)(11 19 17 15 13)(12 20 18 16 14)(21 29 27 25 23)(22 30 28 26 24)(31 33 35 37 39)(32 34 36 38 40)(41 45 49 43 47)(42 46 50 44 48)(51 57 53 59 55)(52 58 54 60 56)(61 67 63 69 65)(62 68 64 70 66)(71 75 79 73 77)(72 76 80 74 78)
(1 52 12 42 6 57 17 47)(2 53 13 43 7 58 18 48)(3 54 14 44 8 59 19 49)(4 55 15 45 9 60 20 50)(5 56 16 46 10 51 11 41)(21 76 36 66 26 71 31 61)(22 77 37 67 27 72 32 62)(23 78 38 68 28 73 33 63)(24 79 39 69 29 74 34 64)(25 80 40 70 30 75 35 65)```

`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), (1,22)(2,21)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,40)(20,39)(41,63)(42,62)(43,61)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,73)(52,72)(53,71)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14)(21,29,27,25,23)(22,30,28,26,24)(31,33,35,37,39)(32,34,36,38,40)(41,45,49,43,47)(42,46,50,44,48)(51,57,53,59,55)(52,58,54,60,56)(61,67,63,69,65)(62,68,64,70,66)(71,75,79,73,77)(72,76,80,74,78), (1,52,12,42,6,57,17,47)(2,53,13,43,7,58,18,48)(3,54,14,44,8,59,19,49)(4,55,15,45,9,60,20,50)(5,56,16,46,10,51,11,41)(21,76,36,66,26,71,31,61)(22,77,37,67,27,72,32,62)(23,78,38,68,28,73,33,63)(24,79,39,69,29,74,34,64)(25,80,40,70,30,75,35,65)>;`

`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), (1,22)(2,21)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,40)(20,39)(41,63)(42,62)(43,61)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,73)(52,72)(53,71)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14)(21,29,27,25,23)(22,30,28,26,24)(31,33,35,37,39)(32,34,36,38,40)(41,45,49,43,47)(42,46,50,44,48)(51,57,53,59,55)(52,58,54,60,56)(61,67,63,69,65)(62,68,64,70,66)(71,75,79,73,77)(72,76,80,74,78), (1,52,12,42,6,57,17,47)(2,53,13,43,7,58,18,48)(3,54,14,44,8,59,19,49)(4,55,15,45,9,60,20,50)(5,56,16,46,10,51,11,41)(21,76,36,66,26,71,31,61)(22,77,37,67,27,72,32,62)(23,78,38,68,28,73,33,63)(24,79,39,69,29,74,34,64)(25,80,40,70,30,75,35,65) );`

`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)], [(1,22),(2,21),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,40),(20,39),(41,63),(42,62),(43,61),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,73),(52,72),(53,71),(54,80),(55,79),(56,78),(57,77),(58,76),(59,75),(60,74)], [(1,3,5,7,9),(2,4,6,8,10),(11,19,17,15,13),(12,20,18,16,14),(21,29,27,25,23),(22,30,28,26,24),(31,33,35,37,39),(32,34,36,38,40),(41,45,49,43,47),(42,46,50,44,48),(51,57,53,59,55),(52,58,54,60,56),(61,67,63,69,65),(62,68,64,70,66),(71,75,79,73,77),(72,76,80,74,78)], [(1,52,12,42,6,57,17,47),(2,53,13,43,7,58,18,48),(3,54,14,44,8,59,19,49),(4,55,15,45,9,60,20,50),(5,56,16,46,10,51,11,41),(21,76,36,66,26,71,31,61),(22,77,37,67,27,72,32,62),(23,78,38,68,28,73,33,63),(24,79,39,69,29,74,34,64),(25,80,40,70,30,75,35,65)]])`

34 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 5E 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 20A 20B 20C 20D 40A ··· 40H order 1 2 2 4 4 4 5 5 5 5 5 8 8 8 8 10 10 10 10 10 10 10 20 20 20 20 40 ··· 40 size 1 1 10 5 5 50 2 2 4 8 8 10 10 50 50 2 2 4 8 8 20 20 10 10 10 10 10 ··· 10

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 8 type + + + + + + + + - + - image C1 C2 C2 C2 C4 C4 D5 M4(2) D10 C4×D5 C8⋊D5 F5 C2×F5 C22.F5 D5×F5 D10.F5 kernel D10.F5 C5×C5⋊C8 C52⋊3C8 D5×Dic5 C52⋊6C4 D5×C10 C5⋊C8 C52 Dic5 C10 C5 D10 C10 C5 C2 C1 # reps 1 1 1 1 2 2 2 2 2 4 8 1 1 2 2 2

Matrix representation of D10.F5 in GL6(𝔽41)

 34 7 0 0 0 0 34 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 11 0 0 0 0 30 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 40 0 0 1 0 0 40 0 0 0 0 0 40 0 1 0
,
 10 3 0 0 0 0 38 31 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(6,GF(41))| [34,34,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,30,0,0,0,0,11,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0],[10,38,0,0,0,0,3,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0] >;`

D10.F5 in GAP, Magma, Sage, TeX

`D_{10}.F_5`
`% in TeX`

`G:=Group("D10.F5");`
`// GroupNames label`

`G:=SmallGroup(400,122);`
`// by ID`

`G=gap.SmallGroup(400,122);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,970,5765,2897]);`
`// 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,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^3>;`
`// generators/relations`

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