Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | A4⋊GL(2,𝔽3) |
| Fitting | C22×Q8 ⋊ C3⋊D3 |
| derived | A4×SL(2,𝔽3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4⋊S4 |
| commutator, cosocle, hypocenter | A4×SL(2,𝔽3) | C2 |
| Fitting | C22×Q8 | C3⋊D3 |
| socle | C23 | C3⋊S4 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
144 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2,3^2,2^5|f^2=h,g^2=h,b^a=b^2,c^a=c^2,d^a=e,e^a=d,f^a=gh,g^a=fh,f^b=fg,g^b=f,d^c=de,e^c=d,g^f=gh23 conjugacy classes — view the full character table.
| maximal subgroup of | #11923445, #11923447, #11923448, … |
| maximal normal subgroup of | #11923445, #11923447, #11923448, … |
| maximal quotient of | #11921464, #11921466, A4⋊#859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8280);