Finite Group DatabasebetaFinite Group Databasebeta| order | 30324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C19 |
| direct product | C22×C7⋊#11762246 |
| Fitting, socle, Hall | C22×C7×C192 ⋊ C3 |
| derived, Hall | C7×C192 ⋊ C22×C3 |
| Hall | C22×C192 ⋊ ΓL(1,𝔽8) |
| Sylow | C192 ⋊ C22×ΓL(1,𝔽8) |
| Hall | C22×C7 ⋊ #11762246 |
| Sylow | C7 ⋊ C22×#11762246 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C7⋊#11762246 |
| commutator, cosocle, hypocenter | C7×C192 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7×C192 | C3 |
| LCS | G ▷ C7×C192 | |
| UCS | C1 ◁ C22 | |
140 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1596 |
3380 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30324, 309);