Finite Group DatabasebetaFinite Group Databasebeta| order | 125 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | Aut(G) = #92539 (order 500) · Inn(G) = C52 · Out(G) = AGL(1,𝔽5) |
| holomorph | G ⋊ Aut(G) = #72229822 |
| homology | Abelianization (H1) = C52 · Schur multiplier (H2) = C1 |
| extraspecial | 5−1+2 |
| unambiguous semidirect product | C25⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C5 | C52 |
| prime radical | C52 | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5 ◁ G | |
| LECS | G ▷ C5 ▷ C1 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4,8,15,3,6,10,17,22,7,13,19,16,23,14,12,9,18,24,21,5,11,20,25);(2,5,12,13,6)(4,9,10,11,19)(8,16,20,17,18)(15,22,23,24,25)Polycyclic presentation:
5^3|a^5=c,b^a=bc29 conjugacy classes — view the full character table.
| Fitting subgroup of | D25⋊C5, AGL(1,ℤ/25) |
| maximal subgroup of | D25⋊C5, C2×5−1+2, C3×5−1+2, … |
| maximal normal subgroup of | D25⋊C5, C2×5−1+2, C3×5−1+2, … |
| maximal quotient of | C2×5−1+2, C3×5−1+2, C52⋊C25, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(125, 4);