Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #89091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| direct product | C2×#89091 |
| Fitting, socle | C2×C33 ⋊ SD16 |
| Sylow | C33 ⋊ C2×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #89091 |
| commutator, cosocle | C3×#157 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C33 | SD16 |
| prime radical | C2×C3×GO+(4,𝔽2) | C2 |
| hypocenter | C33 | C2×SD16 |
| LCS | G ▷ C3×#157 ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
178 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | #89091⋊C4, #13180139, #13180142, … |
| maximal normal subgroup of | #89091⋊C4, #13180139, #13180142, … |
| maximal quotient of | #13180396, #13180397, #13180398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4683);