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G = C25⋊D5order 250 = 2·53

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

metabelian, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C5×C25 — C25⋊D5
C1C5C52C5×C25 — C25⋊D5
C5×C25 — C25⋊D5
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
5C5⋊D5

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···5L25A···25AX
order125···525···25
size11252···22···2

64 irreducible representations

dim11222
type+++++
imageC1C2D5D5D25
kernelC25⋊D5C5×C25C25C52C5
# reps1110250

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

321100
907200
004393
00869
,
2210000
1000
0001
0010022
,
2210000
797900
000100
001000
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

Subgroup lattice of C25⋊D5 in TeX

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