Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #272 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C42×#272 |
| Fitting | C42×H(3,𝔽3) ⋊ C2 |
| derived, Sylow | H(3,𝔽3) ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C42 | C3⋊D3 |
| commutator, hypocenter | H(3,𝔽3) | C2×C42 |
| Frattini, socle | C22×C3 | C22×C3⋊D3 |
| Fitting | C42×H(3,𝔽3) | C2 |
| prime radical | C22×#272 | C22 |
| cosocle | C22×H(3,𝔽3) | C23 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C42 | |
594 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | #13136885, #13137303, #13137306, … |
| maximal normal subgroup of | #13136885, #13137303, #13137306, … |
| maximal quotient of | #13143708, #13143711, #13143712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1684);