Copied to
clipboard

## G = C25⋊D5order 250 = 2·53

### The semidirect product of C25 and D5 acting via D5/C5=C2

Aliases: C25⋊D5, C5⋊D25, C52.3D5, (C5×C25)⋊3C2, C5.(C5⋊D5), SmallGroup(250,7)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C25⋊D5
G = < a,b,c | a25=b5=c2=1, ab=ba, cac=a-1, cbc=b-1 >

125C2
25D5
25D5
25D5
25D5
25D5
25D5
5D25
5D25
5D25
5D25
5D25

Smallest permutation representation of C25⋊D5
On 125 points
Generators in S125
```(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)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125)
(1 52 43 91 118)(2 53 44 92 119)(3 54 45 93 120)(4 55 46 94 121)(5 56 47 95 122)(6 57 48 96 123)(7 58 49 97 124)(8 59 50 98 125)(9 60 26 99 101)(10 61 27 100 102)(11 62 28 76 103)(12 63 29 77 104)(13 64 30 78 105)(14 65 31 79 106)(15 66 32 80 107)(16 67 33 81 108)(17 68 34 82 109)(18 69 35 83 110)(19 70 36 84 111)(20 71 37 85 112)(21 72 38 86 113)(22 73 39 87 114)(23 74 40 88 115)(24 75 41 89 116)(25 51 42 90 117)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 112)(8 111)(9 110)(10 109)(11 108)(12 107)(13 106)(14 105)(15 104)(16 103)(17 102)(18 101)(19 125)(20 124)(21 123)(22 122)(23 121)(24 120)(25 119)(26 35)(27 34)(28 33)(29 32)(30 31)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(51 92)(52 91)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 100)(69 99)(70 98)(71 97)(72 96)(73 95)(74 94)(75 93)```

`G:=sub<Sym(125)| (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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125), (1,52,43,91,118)(2,53,44,92,119)(3,54,45,93,120)(4,55,46,94,121)(5,56,47,95,122)(6,57,48,96,123)(7,58,49,97,124)(8,59,50,98,125)(9,60,26,99,101)(10,61,27,100,102)(11,62,28,76,103)(12,63,29,77,104)(13,64,30,78,105)(14,65,31,79,106)(15,66,32,80,107)(16,67,33,81,108)(17,68,34,82,109)(18,69,35,83,110)(19,70,36,84,111)(20,71,37,85,112)(21,72,38,86,113)(22,73,39,87,114)(23,74,40,88,115)(24,75,41,89,116)(25,51,42,90,117), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,112)(8,111)(9,110)(10,109)(11,108)(12,107)(13,106)(14,105)(15,104)(16,103)(17,102)(18,101)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,35)(27,34)(28,33)(29,32)(30,31)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)>;`

`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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125), (1,52,43,91,118)(2,53,44,92,119)(3,54,45,93,120)(4,55,46,94,121)(5,56,47,95,122)(6,57,48,96,123)(7,58,49,97,124)(8,59,50,98,125)(9,60,26,99,101)(10,61,27,100,102)(11,62,28,76,103)(12,63,29,77,104)(13,64,30,78,105)(14,65,31,79,106)(15,66,32,80,107)(16,67,33,81,108)(17,68,34,82,109)(18,69,35,83,110)(19,70,36,84,111)(20,71,37,85,112)(21,72,38,86,113)(22,73,39,87,114)(23,74,40,88,115)(24,75,41,89,116)(25,51,42,90,117), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,112)(8,111)(9,110)(10,109)(11,108)(12,107)(13,106)(14,105)(15,104)(16,103)(17,102)(18,101)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,35)(27,34)(28,33)(29,32)(30,31)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93) );`

`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),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125)], [(1,52,43,91,118),(2,53,44,92,119),(3,54,45,93,120),(4,55,46,94,121),(5,56,47,95,122),(6,57,48,96,123),(7,58,49,97,124),(8,59,50,98,125),(9,60,26,99,101),(10,61,27,100,102),(11,62,28,76,103),(12,63,29,77,104),(13,64,30,78,105),(14,65,31,79,106),(15,66,32,80,107),(16,67,33,81,108),(17,68,34,82,109),(18,69,35,83,110),(19,70,36,84,111),(20,71,37,85,112),(21,72,38,86,113),(22,73,39,87,114),(23,74,40,88,115),(24,75,41,89,116),(25,51,42,90,117)], [(1,118),(2,117),(3,116),(4,115),(5,114),(6,113),(7,112),(8,111),(9,110),(10,109),(11,108),(12,107),(13,106),(14,105),(15,104),(16,103),(17,102),(18,101),(19,125),(20,124),(21,123),(22,122),(23,121),(24,120),(25,119),(26,35),(27,34),(28,33),(29,32),(30,31),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(51,92),(52,91),(53,90),(54,89),(55,88),(56,87),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,100),(69,99),(70,98),(71,97),(72,96),(73,95),(74,94),(75,93)]])`

C25⋊D5 is a maximal subgroup of   C25⋊F5  C252F5  D5×D25
C25⋊D5 is a maximal quotient of   C50.D5

64 conjugacy classes

 class 1 2 5A ··· 5L 25A ··· 25AX order 1 2 5 ··· 5 25 ··· 25 size 1 125 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D5 D5 D25 kernel C25⋊D5 C5×C25 C25 C52 C5 # reps 1 1 10 2 50

Matrix representation of C25⋊D5 in GL4(𝔽101) generated by

 32 11 0 0 90 72 0 0 0 0 43 93 0 0 8 69
,
 22 100 0 0 1 0 0 0 0 0 0 1 0 0 100 22
,
 22 100 0 0 79 79 0 0 0 0 0 100 0 0 100 0
`G:=sub<GL(4,GF(101))| [32,90,0,0,11,72,0,0,0,0,43,8,0,0,93,69],[22,1,0,0,100,0,0,0,0,0,0,100,0,0,1,22],[22,79,0,0,100,79,0,0,0,0,0,100,0,0,100,0] >;`

C25⋊D5 in GAP, Magma, Sage, TeX

`C_{25}\rtimes D_5`
`% in TeX`

`G:=Group("C25:D5");`
`// GroupNames label`

`G:=SmallGroup(250,7);`
`// by ID`

`G=gap.SmallGroup(250,7);`
`# by ID`

`G:=PCGroup([4,-2,-5,-5,-5,465,805,482,3203]);`
`// Polycyclic`

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

Export

׿
×
𝔽