Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #272 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×#4208 |
| Fitting | C4×#737 ⋊ C2 |
| Sylow | #737 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | C3⋊D3 |
| commutator, hypocenter | H(3,𝔽3) | C2×C3×C4 |
| Frattini, socle | C2×C3 | C2×C3×C3⋊D3 |
| Fitting | C4×#737 | C2 |
| prime radical | C2×#272 | C2×C3 |
| cosocle | C2×H(3,𝔽3) | C22×C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C4×C9 | |
160 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | #13047758, #13047759, C8×#4208, … |
| maximal normal subgroup of | #13047758, #13047759, C8×#4208, … |
| maximal quotient of | C8×#4208, #13047916, C4×#66410, … |
| automorphism group of | C5×C27⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 488);