Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊#157 |
| commutator, socle | C2×C33 | C22×C4 |
| Fitting | C33×Q8 | C4 |
| prime radical | C3×#3832 | C2 |
| cosocle | C3×C3⋊Dic3 | C23 |
| hypercenter | Q8 | C3⋊#157 |
| hypocenter | C33 | #131 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
228 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^5,3^3|a^2=d,b^2=e,c^2=e,d^2=e,f^a=g,g^a=f^2,h^a=h^2,c^b=ce,f^d=f^2,g^d=g^260 conjugacy classes — view the full character table.
| maximal subgroup of | #13166567, #13166568, #13166569, … |
| maximal normal subgroup of | #13166567, #13166568, #13166569, … |
| maximal quotient of | #13173345, #13173346, #13173347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4484);