Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | A4⋊S4 |
| commutator, prime radical, cosocle, hypocenter | Ω+(4,𝔽3) | C2 |
| Fitting | H(5,𝔽2) | C3⋊D3 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2 | |
107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2,3^2,2^5|a^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=b^2,c^a=c^2,d^a=eh,e^a=dh,f^a=gh,g^a=fh,f^b=fg,g^b=f,d^c=de,e^c=d,e^d=eh,g^f=gh20 conjugacy classes — view the full character table.
| maximal subgroup of | #11923440, #11923451, #11923453, … |
| maximal normal subgroup of | #11923440, #11923451, #11923453, … |
| maximal quotient of | #11921464, #11921471, C2×#10596399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8281);