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 | C22×C32×SL(2,𝔽3) |
| Fitting | C22×C32×Q8 ⋊ C3 |
| derived | Q8 ⋊ C22×C33 |
| Sylow | C22×Q8 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C32 | A4 |
| commutator, cosocle, hypocenter | Q8 | C22×C33 |
| Frattini | C2 | C22×C32×A4 |
| Fitting | C22×C32×Q8 | C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23×C32 | |
702 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C32×#859, C32×#5419, C32×#5420, … |
| maximal normal subgroup of | C2×C32×#859, C32×#5419, C32×#5420, … |
| maximal quotient of | C2×C32×C4×SL(2,𝔽3), C32×D4×SL(2,𝔽3), C32×Q8×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4603);