Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of C3⋊(C5⋊(C5⋊S4)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3×C5 |
| unambiguous semidirect product | C3⋊(C5⋊(C5⋊S4)) |
| unambiguous semidirect product | C5⋊(C3⋊(C5⋊S4)) |
| unambiguous semidirect product | A4⋊(C3⋊(C5⋊D5)) |
| Fitting, socle | C22×C3×C52 ⋊ D3 |
| derived | C3×C52×A4 ⋊ C2 |
| Sylow | C52 ⋊ C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C52×A4 | C2 |
| Fitting, socle | C22×C3×C52 | D3 |
| LCS | G ▷ C3×C52×A4 | |
| UCS | C1 | |
240 conjugacy classes of subgroups.
| transitive degree | μt(G) = 300 |
Polycyclic presentation:
2,3^2,5^2,2^2|b^a=b^2,c^a=c^2,d^a=d^4,e^a=e^4,f^a=fg,f^b=g,g^b=fg153 conjugacy classes — view the full character table.
| maximal subgroup of | #72230199, C5⋊D5×C3⋊S4, D15×C5⋊S4 |
| maximal normal subgroup of | C5⋊D5×C3⋊S4, D15×C5⋊S4 |
| maximal quotient of | #72230199 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 690);