Finite Group DatabasebetaFinite Group Databasebeta| order | 20700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D75⋊D69 |
| Fitting, derived, Hall | C32×C23×C25 ⋊ C22 |
| Hall | C23×C25 ⋊ D32 |
| Hall | C32×C25 ⋊ C2×D23 |
| Hall | C32×C23 ⋊ C2×D25 |
| Sylow | C25 ⋊ D3×D69 |
| Sylow | C23 ⋊ (C3⋊D3)⋊D25 |
| Sylow | C32 ⋊ D23×D25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C32×C23×C25 | C22 |
| Frattini | C5 | D15⋊D69 |
| socle | C32×C5×C23 | C2×D5 |
| LCS | G ▷ C32×C23×C25 | |
| UCS | C1 | |
132 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 34 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:
2^2,3^2,5,23,5|e^5=g^4,c^a=c^2,d^a=d^2,f^a=f^22,d^b=d^2,e^b=e^4g,g^b=g^41368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20700, 87);