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G = C125⋊C4order 500 = 22·53

The semidirect product of C125 and C4 acting faithfully

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

C1C125 — C125⋊C4
C1C5C25C125D125 — C125⋊C4
C125 — C125⋊C4
C1

Generators and relations for C125⋊C4
 G = < a,b | a125=b4=1, bab-1=a68 >

125C2
125C4
25D5
25F5
5D25
5C25⋊C4

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 4A4B 5 25A···25E125A···125Y
order1244525···25125···125
size112512512544···44···4

35 irreducible representations

dim111444
type+++++
imageC1C2C4F5C25⋊C4C125⋊C4
kernelC125⋊C4D125C125C25C5C1
# reps1121525

Matrix representation of C125⋊C4 in GL4(𝔽3001) generated by

2804202489425
297627791999869
2132210719101130
18711002977780
,
156917441069491
2326174812572826
251010781253578
175250119231432
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

Export

Subgroup lattice of C125⋊C4 in TeX

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