Finite Group DatabasebetaFinite Group Databasebeta| order | 2310 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)×D11 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×ΓL(1,𝔽8)×D11 |
| unambiguous semidirect product | (C5×C7×D11)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | ΓL(1,𝔽8)×D11 |
| commutator, cosocle, hypocenter | C7×C11 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C5×C7×C11 | C2×C3 |
| LCS | G ▷ C7×C11 | |
| UCS | C1 ◁ C5 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 385 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5)(7,8,9)(10,12,11)(13,14,15,16,17,18,19,20,21,22,23);(6,7,8,10,9,11,12)(14,23)(15,22)(16,21)(17,20)(18,19)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2310, 27);