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 | C22×Q8⋊C27 |
| derived | Q8 ⋊ C22×C27 |
| Sylow | C22×Q8 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C9 | A4 |
| commutator, hypocenter | Q8 | C22×C27 |
| Frattini | C2×C9 | C22×A4 |
| Fitting | C22×Q8×C9 | C3 |
| socle, prime radical | C23×C3 | C22⋊C9 |
| cosocle | Q8×C9 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23×C9 | |
141 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11720108, #13136157, #13136158, … |
| maximal normal subgroup of | C2×#11720108, #13136157, #13136158, … |
| maximal quotient of | C2×C4×Q8⋊C27, D4×Q8⋊C27, Q8×Q8⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 624);