Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22⋊C4×C27 ⋊ C2 |
| Sylow | C27 ⋊ #121 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D27 |
| commutator, cosocle | C22×C27 | C23 |
| Frattini | C22×C9 | C22×D3 |
| Fitting | C22⋊C4×C27 | C2 |
| socle | C22×C3 | C22×D9 |
| hypercenter | C22⋊C4 | D27 |
| hypocenter | C27 | #121 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
188 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 51 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=d,f^3=g,g^3=h,c^a=cde,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=ce144 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11720133, #13136229, #13136232, … |
| maximal normal subgroup of | C2×#11720133, #13136229, #13136232, … |
| maximal quotient of | #13134152, #13134157, #13134163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 100);