Finite Group DatabasebetaFinite Group Databasebeta| order | 1323 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C7 · Schur multiplier (H2) = C33 |
| direct product | C32×C7×ΓL(1,𝔽8) |
| unambiguous semidirect product | C7⋊(C33×C7) |
| unambiguous semidirect product | (C3×C7)⋊(C32×C7) |
| Fitting, socle | C32×C72 ⋊ C3 |
| derived | C7 ⋊ C33×C7 |
| Sylow | C72 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C7 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C33×C7 |
| Frattini | C1 | G |
| Fitting, socle | C32×C72 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C32×C7 | |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 189 |
315 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×ΓL(1,𝔽8)2, C32×ΓL(1,𝔽8)⋊D7, C7×C3⋊D3×ΓL(1,𝔽8), … |
| maximal normal subgroup of | C32×ΓL(1,𝔽8)2, C32×ΓL(1,𝔽8)⋊D7, C7×C3⋊D3×ΓL(1,𝔽8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1323, 48);