Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C3×C4 |
No aliases for this group.
| Sylow | C33 ⋊ #117 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×C3⋊D3 |
| commutator | C2×C33 | C22×C4 |
| Fitting, prime radical | C23×C33 | C22 |
| socle, cosocle | C22×C33 | C23 |
| hypercenter | C23 | D3×C3⋊D3 |
| hypocenter | C33 | #117 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
592 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3^3|a^2=e,b^2=d,h^a=h^2,c^b=ce,f^b=f^2,g^b=g^2126 conjugacy classes — view the full character table.
| maximal subgroup of | #13156979, #13159067, #13160900, … |
| maximal normal subgroup of | #13156979, #13159067, #13160900, … |
| maximal quotient of | #13160260, #13160284, #13160285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3561);