Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #88935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#88935 |
| Sylow | C33 ⋊ C2×D8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D32⋊D3 |
| commutator, cosocle | C3×C3⋊Dic3 | C23 |
| Frattini | C2 | C2×D32⋊D3 |
| Fitting, socle | C22×C33 | D4 |
| hypocenter | C33 | C2×D8 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
324 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #13165493, #13165495, #13165524, … |
| maximal normal subgroup of | #13165493, #13165495, #13165524, … |
| maximal quotient of | #13166982, #13166983, #13166984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4288);