Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #6290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C4×#6293 |
| Sylow | C33 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D3×C3⋊D3 |
| commutator | C2×C33 | C22×C4 |
| Frattini | C22 | C2×D3×C3⋊D3 |
| Fitting | C2×C33×C4 | C22 |
| socle, prime radical, cosocle | C22×C33 | C23 |
| hypocenter | C33 | C4×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
540 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 150 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 | #13155762, #13155763, #13155773, … |
| maximal normal subgroup of | #13155762, #13155763, #13155773, … |
| maximal quotient of | C8×#6293, #13159713, #13159717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3500);