Copied to
clipboard

## G = C5×C25order 125 = 53

### Abelian group of type [5,25]

Aliases: C5×C25, SmallGroup(125,2)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C5×C25
 Chief series C1 — C5 — C52 — C5×C25
 Lower central C1 — C5×C25
 Upper central C1 — C5×C25
 Jennings C1 — C5 — C5 — C5 — C5 — 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 ··· 5X 25A ··· 25CV order 1 5 ··· 5 25 ··· 25 size 1 1 ··· 1 1 ··· 1

125 irreducible representations

 dim 1 1 1 1 type + image C1 C5 C5 C25 kernel C5×C25 C25 C52 C5 # reps 1 20 4 100

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

 95 0 0 87
,
 92 0 0 25
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

׿
×
𝔽