Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3×C9 ⋊ #127 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D9 |
| commutator, cosocle | C22×C3×C9 | C23 |
| Frattini | C22×C3 | C2×D32 |
| Fitting | C23×C3×C9 | C22 |
| socle | C22×C32 | C22×D3 |
| hypercenter | C23 | D3×D9 |
| hypocenter | C3×C9 | #127 |
| LCS | G ▷ C22×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C22 ◁ C23 | |
378 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3^3|f^3=h,b^a=be,c^a=cd,g^a=g^2,c^b=cde,f^b=f^2h^2,h^b=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | #13147931, #13147932, #13147933, … |
| maximal normal subgroup of | #13147931, #13147932, #13147933, … |
| maximal quotient of | #13139537, #13139593, #13139670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1256);