Finite Group DatabasebetaFinite Group Databasebeta| order | 6300 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C52 · Schur multiplier (H2) = C2×C3×C5 |
| direct product | C22×C3×C52×ΓL(1,𝔽8) |
| Fitting, socle | C22×C3×C52×C7 ⋊ C3 |
| derived, Sylow | C7 ⋊ C22×C32×C52 |
| Hall | C22×C52×C7 ⋊ C32 |
| Hall | C52×C7 ⋊ C22×C32 |
| Hall | C22×C7 ⋊ C32×C52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C52 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C22×C32×C52 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C52×C7 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C3×C52 | |
480 conjugacy classes of subgroups.
1500 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6300, 551);