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G = D5.D25order 500 = 22·53

The non-split extension by D5 of D25 acting via D25/C25=C2

Aliases: D5.D25, C5⋊Dic25, C253F5, C52.2Dic5, (C5×C25)⋊2C4, (C5×D5).2D5, (D5×C25).2C2, C5.3(D5.D5), SmallGroup(500,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C25 — D5.D25
 Chief series C1 — C5 — C52 — C5×C25 — D5×C25 — D5.D25
 Lower central C5×C25 — D5.D25
 Upper central C1

Generators and relations for D5.D25
G = < a,b,c,d | a5=b2=c25=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a2, bc=cb, dbd-1=ab, dcd-1=c-1 >

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

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 2 2 2 2 4 4 4 type + + + - + - + image C1 C2 C4 D5 Dic5 D25 Dic25 F5 D5.D5 D5.D25 kernel D5.D25 D5×C25 C5×C25 C5×D5 C52 D5 C5 C25 C5 C1 # reps 1 1 2 2 2 10 10 1 4 20

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

 87 0 0 0 0 36 0 0 0 0 84 0 0 0 0 95
,
 0 36 0 0 87 0 0 0 0 0 0 95 0 0 84 0
,
 24 0 0 0 0 24 0 0 0 0 80 0 0 0 0 80
,
 0 0 80 0 0 0 0 80 0 24 0 0 24 0 0 0
`G:=sub<GL(4,GF(101))| [87,0,0,0,0,36,0,0,0,0,84,0,0,0,0,95],[0,87,0,0,36,0,0,0,0,0,0,84,0,0,95,0],[24,0,0,0,0,24,0,0,0,0,80,0,0,0,0,80],[0,0,0,24,0,0,24,0,80,0,0,0,0,80,0,0] >;`

D5.D25 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(500,19);`
`// by ID`

`G=gap.SmallGroup(500,19);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,10,1742,1512,1203,808,10004]);`
`// Polycyclic`

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

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