Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | AΣL(1,𝔽27) ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4174⋊C2 |
| commutator, Fitting, cosocle | C2×AΣL(1,𝔽27) | C22 |
| Frattini | C2×C32 | D32 |
| socle | C2×C3 | ((C3⋊D3)⋊C3)⋊C2 |
| hypocenter | AΣL(1,𝔽27) | D4 |
| LCS | G ▷ C2×AΣL(1,𝔽27) ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C2 | |
134 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 18 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:
2^3,3^4|b^a=bc,e^a=e^2,f^a=f^2g,d^b=d^2,f^b=f^2,g^b=g^2,e^d=ef,f^e=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | #13047113, #13047116, #13047119, … |
| maximal normal subgroup of | #13047113, #13047116, #13047119, … |
| maximal quotient of | #13046414, #13046418, #13046420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 69);