Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | Q8×C3⋊#157 |
| Fitting | C33×Q8 ⋊ C4 |
| Sylow | C33 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊#157 |
| commutator, socle | C2×C33 | C22×C4 |
| Fitting | C33×Q8 | C4 |
| prime radical, cosocle | C2×C3×C3⋊D3 | C23 |
| hypercenter | Q8 | C3⋊#157 |
| hypocenter | C33 | C4×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
240 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #13166557, #13166558, #13166561, … |
| maximal normal subgroup of | #13166557, #13166558, #13166561, … |
| maximal quotient of | #13173350, #13173351, #13173352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4485);