Finite Group DatabasebetaFinite Group Databasebeta| order | 20700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C5 |
| unambiguous semidirect product | D69⋊(C3⋊(C5⋊D5)) |
| Fitting, socle, derived, Hall | C32×C52×C23 ⋊ C22 |
| Hall | C52×C23 ⋊ D32 |
| Hall | C32×C52 ⋊ C2×D23 |
| Hall | C32×C23 ⋊ C2×C5⋊D5 |
| Sylow | C52 ⋊ (C3⋊D3)⋊D23 |
| Sylow | C23 ⋊ D3×C3⋊(C5⋊D5) |
| Sylow | C32 ⋊ D23×C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C52×C23 | C22 |
| LCS | G ▷ C32×C52×C23 | |
| UCS | C1 | |
352 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 3450 |
Polycyclic presentation:
2^2,3^2,5^2,23|d^a=d^2,g^a=g^22,c^b=c^2,d^b=d^2,e^b=e^4,f^b=f^41371 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20700, 321);