Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of #127⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #384⋊C9 |
| derived | C42 ⋊ C4×C9 |
| Sylow | #384 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #127⋊C3 |
| commutator, hypocenter | C42 | C4×C9 |
| Frattini, socle, prime radical | C23×C3 | C2×A4 |
| Fitting | C3×#384 | C3 |
| cosocle | C2×C3×C42 | C2×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C2×C3 | |
83 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2,3,2^4|a^2=c,b^3=d,e^2=gh,f^2=g,e^a=egh,f^a=fg,e^b=f,f^b=efh,g^b=h,h^b=gh60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#384⋊C9, C4×#127⋊C9, #2383⋊C9, … |
| maximal normal subgroup of | C2×#384⋊C9, C4×#127⋊C9, #2383⋊C9, … |
| maximal quotient of | #1444⋊C9, #1842⋊C9, #1843⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 196);