Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #338 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | Dic32 |
| commutator | C32×C4 | C42 |
| Frattini, hypercenter | C22×C4 | D32 |
| Fitting, cosocle | C22×C32×C4 | C22 |
| socle | C2×C32 | #95 |
| prime radical | C23×C32 | Q8 |
| hypocenter | C32 | #338 |
| LCS | G ▷ C32×C4 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C22×C4 | |
206 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 84 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:
2^6,3^2|a^2=d,b^2=c,c^2=f,d^2=f,e^2=f,b^a=be,c^a=cf,h^a=h^2,d^b=df,g^b=g^278 conjugacy classes — view the full character table.
| maximal subgroup of | #11771759, #11771776, D3×C3⋊#338, … |
| maximal normal subgroup of | #11771759, #11771776, D3×C3⋊#338, … |
| maximal quotient of | #11770726, #11770727, #11771007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1041);