Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23×C3 |
| unambiguous semidirect product | C3⋊#5934 |
| Fitting | C3×H(5,𝔽2) ⋊ D3 |
| derived | C3×#997 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×#997 | C2 |
| Fitting | C3×H(5,𝔽2) | D3 |
| socle | C2×C3 | #1020 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2 | |
242 conjugacy classes of subgroups.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2,3^2,2^5|b^a=b^2,c^a=c^2,d^a=e,e^a=d,f^a=g,g^a=f,d^b=e,e^b=de,f^b=fg,g^b=f,f^d=fh,g^e=gh24 conjugacy classes — view the full character table.
| maximal subgroup of | #11920731, #11920733, #11920739, … |
| maximal normal subgroup of | #11920731, #11920733, #11920739, … |
| maximal quotient of | C3⋊#85549, C3⋊#85551, C3⋊#85554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8486);