Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C22×AΣL(1,𝔽27) ⋊ C2 |
| Sylow | AΣL(1,𝔽27) ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4175 |
| commutator, cosocle | C2×C33 | C22×C3 |
| Frattini | C2×C32 | C2×C3×D3 |
| Fitting | C22×AΣL(1,𝔽27) | C2 |
| socle | C2×C3 | C2×(C3⋊D3)⋊C3 |
| hypercenter | C22 | #4175 |
| hypocenter | C33 | C3×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C22 | |
138 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3,2,3^3|b^a=bd,e^a=e^2,f^a=f^2,g^a=g^2,e^c=ef,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | #13047190, #13047191, #13047198, … |
| maximal normal subgroup of | #13047190, #13047191, #13047198, … |
| maximal quotient of | #13046635, #13046638, #13046679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 161);