Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C8×C27 ⋊ C2 |
| Sylow | C27 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D108 |
| commutator, cosocle | C4×C27 | C23 |
| Frattini | C4×C9 | C22×D3 |
| Fitting | C2×C8×C27 | C2 |
| socle | C2×C3 | C2×D36 |
| hypercenter | C2×C8 | D27 |
| hypocenter | C27 | #135 |
| LCS | G ▷ C4×C27 ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^5,3^3|b^2=e,c^2=d,d^2=e,f^3=g,g^3=h,c^a=cd,d^a=de,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2222 conjugacy classes — view the full character table.
| maximal subgroup of | #13133984, #13133993, #13134006, … |
| maximal normal subgroup of | #13133984, #13133993, #13134006, … |
| maximal quotient of | #13134188, C4×Dic108, C4×#88360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 121);