Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of #64188⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #64188⋊C2 |
| derived | C22⋊(C22⋊C9) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C9) | C22 |
| Frattini, socle | C22×C3 | C2×S4 |
| Fitting | C3×#120 | D3 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 | |
179 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 179 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.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3^2,2^4|c^3=d^2,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,g^a=gh,e^b=egh,f^b=fg,e^c=f,f^c=ef,g^c=h,h^c=gh24 conjugacy classes — view the full character table.
| maximal subgroup of | #10588305⋊C2, C2×#64188⋊C2, #13146405, … |
| maximal normal subgroup of | #10588305⋊C2, C2×#64188⋊C2, C3×#64188⋊C2, … |
| maximal quotient of | #11919944, #64188⋊C4, #11919946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1380);