Finite Group DatabasebetaFinite Group Databasebeta| order | 15078 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D359 |
| unambiguous semidirect product | C359⋊AGL(1,𝔽7) |
| unambiguous semidirect product | D2513⋊C3 |
| Fitting, socle, derived, Hall | C7×C359 ⋊ C2×C3 |
| Hall | ΓL(1,𝔽8)×C359 ⋊ C2 |
| Hall | D2513 ⋊ C3 |
| Sylow | C359 ⋊ AGL(1,𝔽7) |
| Hall | ΓL(1,𝔽8) ⋊ D359 |
| Sylow | C7 ⋊ C3×D359 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C359 | C2×C3 |
| LCS | G ▷ C7×C359 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2513 |
Polycyclic presentation:
2,3,7,359|c^a=c^6,d^a=d^358,c^b=c^2902 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×D359, C2×ΓL(1,𝔽8)⋊D359 |
| maximal normal subgroup of | AGL(1,𝔽7)×D359, C2×ΓL(1,𝔽8)⋊D359 |
| maximal quotient of | ΓL(1,𝔽8)⋊Dic359, C2×ΓL(1,𝔽8)⋊D359 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15078, 12);