Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#401⋊C3) |
| unambiguous semidirect product | (C3⋊#401)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(#126⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C3×#95 | C2×C3 |
| Frattini | C23 | D3×A4 |
| socle | C23×C3 | C2×A4 |
| prime radical | C3×#95⋊C3 | C2 |
| LCS | G ▷ C3×#95 | |
| UCS | C1 ◁ C2 | |
71 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2^2,3,2^3|a^2=h,c^2=g,d^2=f,c^a=cfg,d^a=dgh,e^a=e^2,c^b=cd,d^b=cfh,f^b=g,g^b=fgh,d^c=dh30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3⋊(#401⋊C3), #73380⋊C3, #73408⋊C3, … |
| maximal normal subgroup of | C2×C3⋊(#401⋊C3), #73380⋊C3, #73408⋊C3, … |
| maximal quotient of | C3⋊(#1443⋊C3), C2×C3⋊(#401⋊C3), #13136799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1998);