Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C32×C22⋊C4 ⋊ C22 |
| Sylow | C32 ⋊ #546 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×C22⋊C4 | C22 |
| hypercenter | C22⋊C4 | D32 |
| hypocenter | C32 | #546 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
715 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^2|c^2=e,c^a=cf,d^a=df,h^a=h^2,d^b=def,g^b=g^2,d^c=df69 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#5550, D3×#5610, #11902412, … |
| maximal normal subgroup of | D3×#5550, D3×#5610, #11902412, … |
| maximal quotient of | #11804795, #11804863, #11805017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 6165);