Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #1113 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#88808 |
| derived, Sylow | C33 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1113 |
| commutator, hypocenter | C33 | C22×C8 |
| Frattini | C4 | C2×#1113 |
| Fitting | C2×C33×C4 | C22 |
| socle | C22×C33 | C2×C4 |
| prime radical | C22×C3×C3⋊D3 | C4 |
| cosocle | C33×C4 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
418 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | #13154989, #13154990, #13154999, … |
| maximal normal subgroup of | #13154989, #13154990, #13154999, … |
| maximal quotient of | #13161474, #13161476, #13161477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3622);