Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C92 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D92 |
| commutator, Fitting, cosocle | C2×C92 | C22 |
| Frattini, socle | C2×C32 | D32 |
| hypocenter | C92 | D4 |
| LCS | G ▷ C2×C92 ▷ C92 | |
| UCS | C1 ◁ C2 | |
93 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|d^3=f,e^3=g,b^a=bc,e^a=e^2g^2,g^a=g^2,d^b=d^2f^2,f^b=f^269 conjugacy classes — view the full character table.
| maximal subgroup of | #13046939, #13046940, #13047074, … |
| maximal normal subgroup of | #13046939, #13046940, #13047074, … |
| maximal quotient of | #13046364, #13046366, #13046369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 48);