Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C33 |
No aliases for this group.
| Fitting | C2×C33×D4 ⋊ C2 |
| Sylow | C33 ⋊ #121 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C23 |
| Fitting | C2×C33×D4 | C2 |
| hypercenter | C2×D4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #121 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
1316 conjugacy classes of subgroups; only the 321 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 321 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^5,3^3|b^2=e,c^a=cd,f^a=f^2,g^a=g^2,h^a=h^2,c^b=ce144 conjugacy classes — view the full character table.
| maximal subgroup of | #13160890, #13160892, #13160902, … |
| maximal normal subgroup of | #13160890, #13160892, #13160902, … |
| maximal quotient of | #13163592, #13163594, #13163597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3946);