Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | A4⋊(#41⋊C3) |
| unambiguous semidirect product | #5919⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×S4 |
| commutator, cosocle, hypocenter | Q8×A4 | C2×C3 |
| Fitting | C22×Q8 | C3×D3 |
| socle | C23 | D3×A4 |
| LCS | G ▷ Q8×A4 | |
| UCS | C1 ◁ C2 | |
112 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 112 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) = 48 |
Polycyclic presentation:
2,3,2^2,3,2^3|a^2=f,c^2=f,d^2=f,e^a=e^2,g^a=h,h^a=g,c^b=d,d^b=cd,d^c=df,g^e=h,h^e=gh35 conjugacy classes — view the full character table.
| maximal subgroup of | #11923443, #11923445, #11923447, … |
| maximal normal subgroup of | #11923443, #11923445, #11923447, … |
| maximal quotient of | #85430⋊C3, #85428⋊C3, SL(2,𝔽3)×SL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8270);