Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C64 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊C64) |
| derived, Sylow | C32 ⋊ C64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | C3⋊D3 |
| commutator, hypocenter | C32 | C64 |
| Fitting, cosocle | C32×C32 | C2 |
| socle, prime radical | C2×C32 | C32 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C32 | |
42 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^2|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^a=g^2,h^a=h^2192 conjugacy classes — view the full character table.
| maximal subgroup of | #11770649, C3⋊D3×C64, C2×C3⋊(C3⋊C64), … |
| maximal normal subgroup of | #11770649, C3⋊D3×C64, C2×C3⋊(C3⋊C64), … |
| maximal quotient of | C3⋊(C3⋊C128), C2×C3⋊(C3⋊C64), C3⋊(C9⋊C64), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 181);