Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of #857⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #857⋊D3 |
| Fitting | C3×C42 ⋊ C2×D3 |
| derived | C3×C42⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C42⋊C3 | C22 |
| Frattini | C22 | D3×S4 |
| Fitting | C3×C42 | C2×D3 |
| socle | C22×C3 | C2×S4 |
| LCS | G ▷ C3×C42⋊C3 | |
| UCS | C1 | |
172 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 172 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.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3^2,2^4|e^2=g,f^2=gh,d^a=d^2,e^a=eg,f^a=fgh,c^b=c^2,e^b=fh,f^b=eh,g^b=gh,e^c=fg,f^c=ef,g^c=gh,h^c=g24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#857⋊D3, D3×#857⋊C2, #857⋊D9, … |
| maximal normal subgroup of | C2×#857⋊D3, D3×#857⋊C2, C3×#857⋊D3 |
| maximal quotient of | #11920468, #11920470, #11920472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5053);