Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of (C22⋊C9)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊C9)⋊C16 |
| derived | C22⋊C9 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | C22⋊C9 | C16 |
| Frattini | C3×C8 | S4 |
| Fitting | C22×C3×C8 | D3 |
| socle, prime radical | C23×C3 | C3⋊C8 |
| cosocle | C8×C22⋊C9 | C2 |
| LCS | G ▷ C22⋊C9 | |
| UCS | C1 ◁ C8 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^4,3^2,2^2|a^2=b,b^2=c,c^2=d,e^3=f^2,e^a=e^2f,f^a=f^2,g^a=h,h^a=g,g^e=h,h^e=gh72 conjugacy classes — view the full character table.
| maximal subgroup of | C16×(C22⋊C9)⋊C2, #11919990, C2×(C22⋊C9)⋊C16, … |
| maximal normal subgroup of | C16×(C22⋊C9)⋊C2, #11919990, C2×(C22⋊C9)⋊C16, … |
| maximal quotient of | (Q8⋊C9)⋊C16, (C22⋊C9)⋊C32, #11919934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 189);