Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #733⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #733⋊C8 |
| derived, Sylow | #733 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #733⋊C2 |
| commutator, hypocenter | #733 | C8 |
| Frattini | C32×C4 | C3⋊D3 |
| Fitting, cosocle | C4×#733 | C2 |
| socle | C2×C3 | #1084 |
| prime radical | C2×C32 | C3⋊Dic3 |
| LCS | G ▷ #733 | |
| UCS | C1 ◁ C4 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,d^3=g^2,e^3=g^2,d^a=d^2g,e^a=e^2g,f^a=fg^2,g^a=g^2,e^d=ef,f^d=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#733⋊C2, #13046760, C2×#733⋊C8, … |
| maximal normal subgroup of | C8×#733⋊C2, #13046760, C2×#733⋊C8, … |
| maximal quotient of | #733⋊C16, C2×#733⋊C8, #6858⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 31);