Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C27 · Schur multiplier (H2) = C2 |
| direct product | C2×#41⋊C27 |
| unambiguous semidirect product | (C2×#41)⋊C27 |
| Sylow | C2×#41 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C9 | A4 |
| commutator, hypocenter | Q8 | C22×C27 |
| Frattini | C2×C9 | C22×A4 |
| Fitting | C2×C9×#41 | C3 |
| socle | C22×C3 | C2×C22⋊C9 |
| prime radical | C2×C3×#41 | C9 |
| cosocle | Q8×C9 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C4×C9 | |
121 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11720109, #13136161, #13136162, … |
| maximal normal subgroup of | C2×#11720109, #13136161, #13136162, … |
| maximal quotient of | C2×C4×Q8⋊C27, C4×#41⋊C27, #548⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 626);