Finite Group DatabasebetaFinite Group Databasebeta| order | 3045 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C29 · Schur multiplier (H2) = C1 |
| Fitting, socle, Hall | C5×C7×C29 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C5×C29 |
| Hall | C7×C29 ⋊ C3×C5 |
| Hall | C5×C7 ⋊ C3×C29 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C29 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C3×C5×C29 |
| Frattini | C1 | G |
| Fitting, socle | C5×C7×C29 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C5×C29 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1015 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5)(7,8,9)(10,12,11)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(6,7,8,10,9,11,12)725 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3045, 3);