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Cayley diagram

C125 #1247

order125
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C4×C25 (order 100) · Inn(G) = C1 · Out(G) = C4×C25
holomorphG ⋊ Aut(G) = #72229807
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC125

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC25C5
socle, prime radicalC5C25
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C25 ▷ C5 ▷ C1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 125
transitive degreeμt(G) = 125

A minimal faithful permutation representation — 1 generator on 125 points:

(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)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

center ofD3×C125, D5×C125, A4×C125, …
derived subgroup ofD125, Dic125, #92517, …
Frattini subgroup ofC625, C2×C625, D625, …
Fitting subgroup ofD125, #92517
maximal subgroup ofD125, C2×C125, C3×C125, …
maximal normal subgroup ofD125, C2×C125, C3×C125, …
maximal quotient ofC2×C125, C3×C125, C625, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(125, 1);