direct product, p-group, abelian, monomial
Aliases: C5×C25, SmallGroup(125,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C5×C25 |
| C1 — C5×C25 |
| C1 — 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 50 123 88 68)(2 26 124 89 69)(3 27 125 90 70)(4 28 101 91 71)(5 29 102 92 72)(6 30 103 93 73)(7 31 104 94 74)(8 32 105 95 75)(9 33 106 96 51)(10 34 107 97 52)(11 35 108 98 53)(12 36 109 99 54)(13 37 110 100 55)(14 38 111 76 56)(15 39 112 77 57)(16 40 113 78 58)(17 41 114 79 59)(18 42 115 80 60)(19 43 116 81 61)(20 44 117 82 62)(21 45 118 83 63)(22 46 119 84 64)(23 47 120 85 65)(24 48 121 86 66)(25 49 122 87 67)
(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)| (1,50,123,88,68)(2,26,124,89,69)(3,27,125,90,70)(4,28,101,91,71)(5,29,102,92,72)(6,30,103,93,73)(7,31,104,94,74)(8,32,105,95,75)(9,33,106,96,51)(10,34,107,97,52)(11,35,108,98,53)(12,36,109,99,54)(13,37,110,100,55)(14,38,111,76,56)(15,39,112,77,57)(16,40,113,78,58)(17,41,114,79,59)(18,42,115,80,60)(19,43,116,81,61)(20,44,117,82,62)(21,45,118,83,63)(22,46,119,84,64)(23,47,120,85,65)(24,48,121,86,66)(25,49,122,87,67), (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( (1,50,123,88,68)(2,26,124,89,69)(3,27,125,90,70)(4,28,101,91,71)(5,29,102,92,72)(6,30,103,93,73)(7,31,104,94,74)(8,32,105,95,75)(9,33,106,96,51)(10,34,107,97,52)(11,35,108,98,53)(12,36,109,99,54)(13,37,110,100,55)(14,38,111,76,56)(15,39,112,77,57)(16,40,113,78,58)(17,41,114,79,59)(18,42,115,80,60)(19,43,116,81,61)(20,44,117,82,62)(21,45,118,83,63)(22,46,119,84,64)(23,47,120,85,65)(24,48,121,86,66)(25,49,122,87,67), (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([[(1,50,123,88,68),(2,26,124,89,69),(3,27,125,90,70),(4,28,101,91,71),(5,29,102,92,72),(6,30,103,93,73),(7,31,104,94,74),(8,32,105,95,75),(9,33,106,96,51),(10,34,107,97,52),(11,35,108,98,53),(12,36,109,99,54),(13,37,110,100,55),(14,38,111,76,56),(15,39,112,77,57),(16,40,113,78,58),(17,41,114,79,59),(18,42,115,80,60),(19,43,116,81,61),(20,44,117,82,62),(21,45,118,83,63),(22,46,119,84,64),(23,47,120,85,65),(24,48,121,86,66),(25,49,122,87,67)], [(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
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