Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #64209 |
| commutator, cosocle | C3×C4×A4 | C22 |
| Frattini, hypercenter | C4 | D3×S4 |
| Fitting | C22×C3×C4 | C2×D3 |
| socle | C23×C3 | #67 |
| hypocenter | C3×A4 | D8 |
| LCS | G ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
182 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^2,2^2|c^2=d,b^a=bc,c^a=cd,f^a=f^2,c^b=cd,e^b=e^2,g^b=h,h^b=g,g^e=h,h^e=gh33 conjugacy classes — view the full character table.
| maximal subgroup of | #11921804, #11921805, #11921806, … |
| maximal normal subgroup of | #11921804, #11921805, #11921806, … |
| maximal quotient of | #11920501, #11920509, #11920543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5065);