Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C33×C8 ⋊ C22 |
| Sylow | C33 ⋊ C2×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×#1113 |
| commutator, socle | C2×C33 | C22×C4 |
| Fitting | C33×C8 | C22 |
| prime radical | C2×#1113 | C4 |
| cosocle | C33×C4 | C23 |
| hypercenter | C8 | #1113 |
| hypocenter | C33 | C2×M16 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ C8 | |
342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^5,3^3|c^2=d,d^2=e,f^a=f^2,h^a=h^2,c^b=ce,f^b=f^2,g^b=g^296 conjugacy classes — view the full character table.
| maximal subgroup of | #13154934, #13155057, #13166911, … |
| maximal normal subgroup of | #13154934, #13155057, #13166911, … |
| maximal quotient of | #13161631, #13161632, #13161633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3612);