Finite Group DatabasebetaFinite Group Databasebeta| order | 15729 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C7×C107 · Schur multiplier (H2) = C1 |
| direct product | C7×ΓL(1,𝔽8)×C107 |
| unambiguous semidirect product | C7⋊(C3×C7×C107) |
| unambiguous semidirect product | (C7×C107)⋊(C3×C7) |
| Fitting, socle, Hall | C72×C107 ⋊ C3 |
| derived | C7 ⋊ C3×C7×C107 |
| Sylow | C72 ⋊ C3×C107 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C107 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C3×C7×C107 |
| Frattini | C1 | G |
| Fitting, socle | C72×C107 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C7×C107 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2247 |
3745 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×ΓL(1,𝔽8)×D107, C7×AGL(1,𝔽7)×C107, C7×ΓL(1,𝔽8)⋊D107, … |
| maximal normal subgroup of | C7×ΓL(1,𝔽8)×D107, C7×AGL(1,𝔽7)×C107, C7×ΓL(1,𝔽8)⋊D107, … |
| maximal quotient of | C2×C7×ΓL(1,𝔽8)×C107 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15729, 5);