Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of GO+(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×GO+(4,𝔽2) |
| unambiguous semidirect product | #157⋊(C2×C32) |
| unambiguous semidirect product | D32⋊(C2×C32) |
| Fitting, socle, Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | GO+(4,𝔽2) |
| commutator, cosocle | C3⋊D3 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C34 | D4 |
| hypocenter | C32 | C32×D4 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C32 | |
226 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
81 conjugacy classes — view the full character table.
| maximal subgroup of | C32×AΓL(1,𝔽9), C3×#89091, #13049796, … |
| maximal normal subgroup of | C32×AΓL(1,𝔽9), C3×#89091, #13049796, … |
| maximal quotient of | C32×D32⋊C4, C32×#3773, C32×#3774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 718);