Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C5 |
| unambiguous semidirect product | C22⋊((C5⋊D5)⋊C5) |
| unambiguous semidirect product | (C22⋊(C5⋊D5))⋊C5 |
| Fitting | C22×H(3,𝔽5) ⋊ C2 |
| Sylow | H(3,𝔽5) ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×(C5⋊D5)⋊C5 |
| commutator, cosocle | C2×C52 | C22×C5 |
| Frattini, socle | C2×C5 | C2×C5×D5 |
| Fitting | C22×H(3,𝔽5) | C2 |
| hypercenter | C22 | (C5⋊D5)⋊C5 |
| hypocenter | C52 | C5×D4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
76 conjugacy classes of subgroups.
| transitive degree | μt(G) = 100 |
Polycyclic presentation:
2^2,5,2,5^2|b^a=bd,e^a=e^4,f^a=f^4,e^c=ef73 conjugacy classes — view the full character table.
| maximal subgroup of | #14522425, #14522426, #14522431, … |
| maximal normal subgroup of | #14522425, #14522426, #14522431, … |
| maximal quotient of | #87806⋊C5, #87808⋊C5, #87810⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 64);