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G = C5×C25order 125 = 53

Abelian group of type [5,25]

direct product, p-group, abelian, monomial

Aliases: C5×C25, SmallGroup(125,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C5×C25
C1C5C52 — C5×C25
C1 — C5×C25
C1 — C5×C25
C1C5C5C5C5 — C5×C25

Generators and relations for C5×C25
 G = < a,b | a5=b25=1, ab=ba >


Smallest permutation representation of C5×C25
Regular action on 125 points
Generators in S125
(1 81 37 105 65)(2 82 38 106 66)(3 83 39 107 67)(4 84 40 108 68)(5 85 41 109 69)(6 86 42 110 70)(7 87 43 111 71)(8 88 44 112 72)(9 89 45 113 73)(10 90 46 114 74)(11 91 47 115 75)(12 92 48 116 51)(13 93 49 117 52)(14 94 50 118 53)(15 95 26 119 54)(16 96 27 120 55)(17 97 28 121 56)(18 98 29 122 57)(19 99 30 123 58)(20 100 31 124 59)(21 76 32 125 60)(22 77 33 101 61)(23 78 34 102 62)(24 79 35 103 63)(25 80 36 104 64)
(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)

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

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

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

C5×C25 is a maximal subgroup of   C25⋊D5

125 conjugacy classes

class 1 5A···5X25A···25CV
order15···525···25
size11···11···1

125 irreducible representations

dim1111
type+
imageC1C5C5C25
kernelC5×C25C25C52C5
# reps1204100

Matrix representation of C5×C25 in GL2(𝔽101) generated by

950
087
,
920
025
G:=sub<GL(2,GF(101))| [95,0,0,87],[92,0,0,25] >;

C5×C25 in GAP, Magma, Sage, TeX

C_5\times C_{25}
% in TeX

G:=Group("C5xC25");
// GroupNames label

G:=SmallGroup(125,2);
// by ID

G=gap.SmallGroup(125,2);
# by ID

G:=PCGroup([3,-5,5,-5,75]);
// Polycyclic

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

Export

Subgroup lattice of C5×C25 in TeX

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