metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C125⋊C4, C25.F5, D125.C2, C5.(C25⋊C4), SmallGroup(500,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C125 — C125⋊C4 |
Generators and relations for C125⋊C4
G = < a,b | a125=b4=1, bab-1=a68 >
Smallest permutation representation of C125⋊C4
►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)
(2 58 125 69)(3 115 124 12)(4 47 123 80)(5 104 122 23)(6 36 121 91)(7 93 120 34)(8 25 119 102)(9 82 118 45)(10 14 117 113)(11 71 116 56)(13 60 114 67)(15 49 112 78)(16 106 111 21)(17 38 110 89)(18 95 109 32)(19 27 108 100)(20 84 107 43)(22 73 105 54)(24 62 103 65)(26 51 101 76)(28 40 99 87)(29 97 98 30)(31 86 96 41)(33 75 94 52)(35 64 92 63)(37 53 90 74)(39 42 88 85)(44 77 83 50)(46 66 81 61)(48 55 79 72)(57 68 70 59)
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), (2,58,125,69)(3,115,124,12)(4,47,123,80)(5,104,122,23)(6,36,121,91)(7,93,120,34)(8,25,119,102)(9,82,118,45)(10,14,117,113)(11,71,116,56)(13,60,114,67)(15,49,112,78)(16,106,111,21)(17,38,110,89)(18,95,109,32)(19,27,108,100)(20,84,107,43)(22,73,105,54)(24,62,103,65)(26,51,101,76)(28,40,99,87)(29,97,98,30)(31,86,96,41)(33,75,94,52)(35,64,92,63)(37,53,90,74)(39,42,88,85)(44,77,83,50)(46,66,81,61)(48,55,79,72)(57,68,70,59)>;
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), (2,58,125,69)(3,115,124,12)(4,47,123,80)(5,104,122,23)(6,36,121,91)(7,93,120,34)(8,25,119,102)(9,82,118,45)(10,14,117,113)(11,71,116,56)(13,60,114,67)(15,49,112,78)(16,106,111,21)(17,38,110,89)(18,95,109,32)(19,27,108,100)(20,84,107,43)(22,73,105,54)(24,62,103,65)(26,51,101,76)(28,40,99,87)(29,97,98,30)(31,86,96,41)(33,75,94,52)(35,64,92,63)(37,53,90,74)(39,42,88,85)(44,77,83,50)(46,66,81,61)(48,55,79,72)(57,68,70,59) );
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)], [(2,58,125,69),(3,115,124,12),(4,47,123,80),(5,104,122,23),(6,36,121,91),(7,93,120,34),(8,25,119,102),(9,82,118,45),(10,14,117,113),(11,71,116,56),(13,60,114,67),(15,49,112,78),(16,106,111,21),(17,38,110,89),(18,95,109,32),(19,27,108,100),(20,84,107,43),(22,73,105,54),(24,62,103,65),(26,51,101,76),(28,40,99,87),(29,97,98,30),(31,86,96,41),(33,75,94,52),(35,64,92,63),(37,53,90,74),(39,42,88,85),(44,77,83,50),(46,66,81,61),(48,55,79,72),(57,68,70,59)]])
35 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 25A | ··· | 25E | 125A | ··· | 125Y |
| order | 1 | 2 | 4 | 4 | 5 | 25 | ··· | 25 | 125 | ··· | 125 |
| size | 1 | 125 | 125 | 125 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | |
| image | C1 | C2 | C4 | F5 | C25⋊C4 | C125⋊C4 |
| kernel | C125⋊C4 | D125 | C125 | C25 | C5 | C1 |
| # reps | 1 | 1 | 2 | 1 | 5 | 25 |
Matrix representation of C125⋊C4 ►in GL4(𝔽3001) generated by
| 2804 | 2024 | 894 | 25 |
| 2976 | 2779 | 1999 | 869 |
| 2132 | 2107 | 1910 | 1130 |
| 1871 | 1002 | 977 | 780 |
| 1569 | 1744 | 1069 | 491 |
| 2326 | 1748 | 1257 | 2826 |
| 2510 | 1078 | 1253 | 578 |
| 175 | 2501 | 1923 | 1432 |
G:=sub<GL(4,GF(3001))| [2804,2976,2132,1871,2024,2779,2107,1002,894,1999,1910,977,25,869,1130,780],[1569,2326,2510,175,1744,1748,1078,2501,1069,1257,1253,1923,491,2826,578,1432] >;
C125⋊C4 in GAP, Magma, Sage, TeX
C_{125}\rtimes C_4 % in TeX
G:=Group("C125:C4"); // GroupNames label
G:=SmallGroup(500,3);
// by ID
G=gap.SmallGroup(500,3);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,10,422,1477,1212,2803,4808,118,5004,5009]);
// Polycyclic
G:=Group<a,b|a^125=b^4=1,b*a*b^-1=a^68>;
// generators/relations
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