Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×S4 |
| commutator | C2×C3×A4 | C2×C4 |
| Fitting, socle | C24×C3 | C2×D3 |
| prime radical, cosocle | C22×C3×A4 | C22 |
| hypocenter | C3×A4 | C4⋊C4 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C22 | |
234 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^4,3^2,2^2|a^2=c,b^2=c,b^a=bd,f^a=f^2,e^b=e^2,g^b=h,h^b=g,g^e=h,h^e=gh48 conjugacy classes — view the full character table.
| maximal subgroup of | #11921884, #11921885, #11921887, … |
| maximal normal subgroup of | #11921884, #11921885, #11921887, … |
| maximal quotient of | #11920616, #11920618, #11920626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5120);