Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C33 |
No aliases for this group.
| Sylow | C33 ⋊ #125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊(C3⋊D3) |
| commutator, socle, prime radical, cosocle | C22×C33 | C23 |
| Fitting | C33×C4⋊C4 | C2 |
| hypercenter | C4⋊C4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #125 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
784 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^5,3^3|a^2=d,b^2=e,c^2=d,c^a=cde,f^a=f^2,g^a=g^2,h^a=h^2,c^b=ce144 conjugacy classes — view the full character table.
| maximal subgroup of | #13159119, #13159120, #13159121, … |
| maximal normal subgroup of | #13159119, #13159120, #13159121, … |
| maximal quotient of | #13163295, #13163302, #13163304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3898);