Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C32×#994 |
| Fitting | C32×H(5,𝔽2) ⋊ C3 |
| Sylow | H(5,𝔽2) ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C22×C33 |
| Frattini | C2 | C22×C32×A4 |
| Fitting | C32×H(5,𝔽2) | C3 |
| hypercenter | C32×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C32 ◁ C32×Q8 | |
452 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
171 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#5428, C32×#5429, C3×#10593662, … |
| maximal normal subgroup of | C32×#5428, C32×#5429, C3×#10593662, … |
| maximal quotient of | C32×#520⋊C3, C32×C4⋊(#41⋊C3), C32×Q8×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4606);