Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C4⋊#4169 |
| Fitting | C4×C32⋊C9 ⋊ C2 |
| Sylow | C32⋊C9 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4169 |
| commutator, cosocle | C2×C3×C9 | C22×C3 |
| Frattini, socle | C2×C32 | C2×C3×D3 |
| Fitting | C4×C32⋊C9 | C2 |
| hypercenter | C4 | #4169 |
| hypocenter | C3×C9 | C3×D4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
132 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3,2,3^3|b^2=d,e^3=g,b^a=bd,e^a=e^2g^2,f^a=f^2,g^a=g^2,e^c=ef75 conjugacy classes — view the full character table.
| maximal subgroup of | #13046378, #13046379, #13046384, … |
| maximal normal subgroup of | #13046378, #13046379, #13046384, … |
| maximal quotient of | #13046530, #13046533, #13046534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 113);