Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #625 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C22×C3×#627 |
| Sylow | C33 ⋊ C22×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D32 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini | C2 | C22×C3×D32 |
| Fitting, socle, prime radical | C23×C33 | C22 |
| hypocenter | C32 | C22×C3×Q8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C23×C3 | |
818 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 818 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | #13156859, #13156860, #13156861, … |
| maximal normal subgroup of | #13156859, #13156860, #13156861, … |
| maximal quotient of | C2×C3×#63869, C2×C3×#63872, C2×C3×C4×#627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4522);