Finite Group DatabasebetaFinite Group Databasebeta| order | 8253 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C9×C131 · Schur multiplier (H2) = C1 |
| direct product | C7⋊C9×C131 |
| unambiguous semidirect product | C7⋊(C9×C131) |
| unambiguous semidirect product | (C7×C131)⋊C9 |
| derived, Sylow | C7 ⋊ C9×C131 |
| Hall | C7×C131 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C131 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C9×C131 |
| Frattini | C3 | ΓL(1,𝔽8)×C131 |
| Fitting, socle | C3×C7×C131 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C131 | |
12 conjugacy classes of subgroups.
1965 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C7⋊C9×C131, C7⋊C9×D131, D7⋊C9×C131, … |
| maximal normal subgroup of | C2×C7⋊C9×C131, C7⋊C9×D131, D7⋊C9×C131, … |
| maximal quotient of | C2×C7⋊C9×C131, C131×C22⋊(C7⋊C9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8253, 2);