Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #6189 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C22×#6189 |
| Sylow | C3×C9 ⋊ C22×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×D9 |
| commutator, cosocle | C2×C3×C9 | C24 |
| Frattini | C2×C3 | C22×D32 |
| Fitting | C23×C3×C9 | C22 |
| socle, prime radical | C23×C32 | C2×D3 |
| hypocenter | C3×C9 | C22×Q8 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C23 | |
522 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | #13139843, #13139844, #13139845, … |
| maximal normal subgroup of | #13139843, #13139844, #13139845, … |
| maximal quotient of | C2×#11720927, C2×#11720937, C2×#11720935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2489);