Finite Group DatabasebetaFinite Group Databasebeta| order | 6250 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, Sylow | #15691060 ⋊ C2 |
| derived | C5×H(3,𝔽5) ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #11929376 |
| commutator, hypocenter | C5×H(3,𝔽5) | C2×C5 |
| Frattini | C53 | C5×D5 |
| Fitting | #15691060 | C2 |
| socle | C52 | (C5⋊D5)⋊C5 |
| LCS | G ▷ C5×H(3,𝔽5) | |
| UCS | C1 ◁ C5 | |
142 conjugacy classes of subgroups.
Polycyclic presentation:
2,5^5|c^a=b^3c^4de^2f^2,d^a=d^4ef,e^a=e^4,f^a=e^2f,c^b=cd,d^b=de,d^c=df112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6250, 81);