Finite Group DatabasebetaFinite Group Databasebeta| order | 30492 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D33⋊(C11⋊AGL(1,𝔽7)) |
| unambiguous semidirect product | (D33⋊D77)⋊C3 |
| Fitting, socle, derived | C3×C7×C112 ⋊ C22×C3 |
| Hall | C3×C112×ΓL(1,𝔽8) ⋊ C22 |
| Hall | C7×C112 ⋊ C2×C3×D3 |
| Sylow | C112 ⋊ D3×AGL(1,𝔽7) |
| Hall | C3×ΓL(1,𝔽8) ⋊ D112 |
| Sylow | C7 ⋊ C3×D33⋊D11 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C112 | C22×C3 |
| LCS | G ▷ C3×C7×C112 | |
| UCS | C1 | |
296 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 462 |
Polycyclic presentation:
2^2,3^2,7,11^2|d^a=d^2,f^a=f^10,e^b=e^6,f^b=f^10,g^b=g^10,e^c=e^2531 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30492, 330);