Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×C32×#41⋊C3 |
| unambiguous semidirect product | (C2×#41)⋊C33 |
| unambiguous semidirect product | (C2×C3×#41)⋊C32 |
| Fitting | C2×C32×#41 ⋊ C3 |
| Sylow | C2×#41 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32×C4 | A4 |
| commutator, cosocle, hypocenter | Q8 | C22×C33 |
| Frattini | C2 | C22×C32×A4 |
| Fitting | C2×C32×#41 | C3 |
| socle | C22×C32 | C2×A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C32×C4 | |
502 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C32×GU(2,𝔽3), C32×#5423, C32×#5424, … |
| maximal normal subgroup of | C2×C32×GU(2,𝔽3), C32×#5423, C32×#5424, … |
| maximal quotient of | C2×C32×C4×SL(2,𝔽3), C32×C4×#41⋊C3, C32×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4605);