Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C33×C4 |
No aliases for this group.
| Sylow | C33 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C23 |
| Fitting, prime radical | C2×C33×D4 | C2 |
| hypercenter | C2×D4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #124 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
1064 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^5,3^3|a^2=e,b^2=d,c^a=cde,f^a=f^2,g^a=g^2,h^a=h^2,c^b=cd144 conjugacy classes — view the full character table.
| maximal subgroup of | #13145683, #13146019, #13161008, … |
| maximal normal subgroup of | #13145683, #13146019, #13161008, … |
| maximal quotient of | #13163585, #13163587, #13163593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3944);