Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C33 |
| direct product | C24×C3⋊(C3⋊D3) |
| Fitting, socle | C24×C33 ⋊ C2 |
| derived, Sylow | C33 ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | C3⋊(C3⋊D3) |
| commutator, cosocle, hypocenter | C33 | C25 |
| Frattini | C1 | G |
| Fitting, socle | C24×C33 | C2 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C24 | |
10472 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | #13161436, #13163596, C22×#88804, … |
| maximal normal subgroup of | #13161436, #13163596, C22×#88804, … |
| maximal quotient of | C23×#6308, C23×C4×C3⋊(C3⋊D3), C23×C4⋊(C3⋊(C3⋊D3)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4724);