Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #88790 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#88790 |
| Sylow | C33 ⋊ C2×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #6291 |
| commutator, cosocle | C33×C4 | C23 |
| Frattini | C4 | C2×D3×C3⋊D3 |
| Fitting | C2×C33×C4 | C22 |
| socle | C22×C33 | D4 |
| hypercenter | C2×C4 | D3×C3⋊D3 |
| hypocenter | C33 | C2×D8 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
672 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
102 conjugacy classes — view the full character table.
| maximal subgroup of | #13155616, #13155619, #13155720, … |
| maximal normal subgroup of | #13155616, #13155619, #13155720, … |
| maximal quotient of | #13158835, #13158841, #13158887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3333);