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## G = D25.D5order 500 = 22·53

### The non-split extension by D25 of D5 acting via D5/C5=C2

Aliases: D25.D5, C25⋊Dic5, C52.3F5, (C5×C25)⋊4C4, C53(C25⋊C4), (C5×D25).2C2, C5.1(D5.D5), SmallGroup(500,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C25 — D25.D5
 Chief series C1 — C5 — C25 — C5×C25 — C5×D25 — D25.D5
 Lower central C5×C25 — D25.D5
 Upper central C1

Generators and relations for D25.D5
G = < a,b,c,d | a25=b2=c5=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c-1 >

Smallest permutation representation of D25.D5
On 100 points
Generators in S100
```(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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)
(1 11 21 6 16)(2 12 22 7 17)(3 13 23 8 18)(4 14 24 9 19)(5 15 25 10 20)(26 41 31 46 36)(27 42 32 47 37)(28 43 33 48 38)(29 44 34 49 39)(30 45 35 50 40)(51 56 61 66 71)(52 57 62 67 72)(53 58 63 68 73)(54 59 64 69 74)(55 60 65 70 75)(76 96 91 86 81)(77 97 92 87 82)(78 98 93 88 83)(79 99 94 89 84)(80 100 95 90 85)
(1 83 26 56)(2 76 50 63)(3 94 49 70)(4 87 48 52)(5 80 47 59)(6 98 46 66)(7 91 45 73)(8 84 44 55)(9 77 43 62)(10 95 42 69)(11 88 41 51)(12 81 40 58)(13 99 39 65)(14 92 38 72)(15 85 37 54)(16 78 36 61)(17 96 35 68)(18 89 34 75)(19 82 33 57)(20 100 32 64)(21 93 31 71)(22 86 30 53)(23 79 29 60)(24 97 28 67)(25 90 27 74)```

`G:=sub<Sym(100)| (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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88), (1,11,21,6,16)(2,12,22,7,17)(3,13,23,8,18)(4,14,24,9,19)(5,15,25,10,20)(26,41,31,46,36)(27,42,32,47,37)(28,43,33,48,38)(29,44,34,49,39)(30,45,35,50,40)(51,56,61,66,71)(52,57,62,67,72)(53,58,63,68,73)(54,59,64,69,74)(55,60,65,70,75)(76,96,91,86,81)(77,97,92,87,82)(78,98,93,88,83)(79,99,94,89,84)(80,100,95,90,85), (1,83,26,56)(2,76,50,63)(3,94,49,70)(4,87,48,52)(5,80,47,59)(6,98,46,66)(7,91,45,73)(8,84,44,55)(9,77,43,62)(10,95,42,69)(11,88,41,51)(12,81,40,58)(13,99,39,65)(14,92,38,72)(15,85,37,54)(16,78,36,61)(17,96,35,68)(18,89,34,75)(19,82,33,57)(20,100,32,64)(21,93,31,71)(22,86,30,53)(23,79,29,60)(24,97,28,67)(25,90,27,74)>;`

`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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88), (1,11,21,6,16)(2,12,22,7,17)(3,13,23,8,18)(4,14,24,9,19)(5,15,25,10,20)(26,41,31,46,36)(27,42,32,47,37)(28,43,33,48,38)(29,44,34,49,39)(30,45,35,50,40)(51,56,61,66,71)(52,57,62,67,72)(53,58,63,68,73)(54,59,64,69,74)(55,60,65,70,75)(76,96,91,86,81)(77,97,92,87,82)(78,98,93,88,83)(79,99,94,89,84)(80,100,95,90,85), (1,83,26,56)(2,76,50,63)(3,94,49,70)(4,87,48,52)(5,80,47,59)(6,98,46,66)(7,91,45,73)(8,84,44,55)(9,77,43,62)(10,95,42,69)(11,88,41,51)(12,81,40,58)(13,99,39,65)(14,92,38,72)(15,85,37,54)(16,78,36,61)(17,96,35,68)(18,89,34,75)(19,82,33,57)(20,100,32,64)(21,93,31,71)(22,86,30,53)(23,79,29,60)(24,97,28,67)(25,90,27,74) );`

`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),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88)], [(1,11,21,6,16),(2,12,22,7,17),(3,13,23,8,18),(4,14,24,9,19),(5,15,25,10,20),(26,41,31,46,36),(27,42,32,47,37),(28,43,33,48,38),(29,44,34,49,39),(30,45,35,50,40),(51,56,61,66,71),(52,57,62,67,72),(53,58,63,68,73),(54,59,64,69,74),(55,60,65,70,75),(76,96,91,86,81),(77,97,92,87,82),(78,98,93,88,83),(79,99,94,89,84),(80,100,95,90,85)], [(1,83,26,56),(2,76,50,63),(3,94,49,70),(4,87,48,52),(5,80,47,59),(6,98,46,66),(7,91,45,73),(8,84,44,55),(9,77,43,62),(10,95,42,69),(11,88,41,51),(12,81,40,58),(13,99,39,65),(14,92,38,72),(15,85,37,54),(16,78,36,61),(17,96,35,68),(18,89,34,75),(19,82,33,57),(20,100,32,64),(21,93,31,71),(22,86,30,53),(23,79,29,60),(24,97,28,67),(25,90,27,74)]])`

38 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C ··· 5G 10A 10B 25A ··· 25Y order 1 2 4 4 5 5 5 ··· 5 10 10 25 ··· 25 size 1 25 125 125 2 2 4 ··· 4 50 50 4 ··· 4

38 irreducible representations

 dim 1 1 1 2 2 4 4 4 4 type + + + - + + image C1 C2 C4 D5 Dic5 F5 C25⋊C4 D5.D5 D25.D5 kernel D25.D5 C5×D25 C5×C25 D25 C25 C52 C5 C5 C1 # reps 1 1 2 2 2 1 5 4 20

Matrix representation of D25.D5 in GL4(𝔽101) generated by

 37 0 0 0 0 71 0 0 0 0 54 0 0 0 0 58
,
 0 71 0 0 37 0 0 0 0 0 0 58 0 0 54 0
,
 87 0 0 0 0 87 0 0 0 0 36 0 0 0 0 36
,
 0 0 36 0 0 0 0 36 0 87 0 0 87 0 0 0
`G:=sub<GL(4,GF(101))| [37,0,0,0,0,71,0,0,0,0,54,0,0,0,0,58],[0,37,0,0,71,0,0,0,0,0,0,54,0,0,58,0],[87,0,0,0,0,87,0,0,0,0,36,0,0,0,0,36],[0,0,0,87,0,0,87,0,36,0,0,0,0,36,0,0] >;`

D25.D5 in GAP, Magma, Sage, TeX

`D_{25}.D_5`
`% in TeX`

`G:=Group("D25.D5");`
`// GroupNames label`

`G:=SmallGroup(500,20);`
`// by ID`

`G=gap.SmallGroup(500,20);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,3203,3808,218,7504,5009]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^25=b^2=c^5=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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