Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of C3⋊#1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23×C3 |
| unambiguous semidirect product | C3⋊#5936 |
| derived | C3×C22⋊A4 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#1020 |
| commutator, hypocenter | C3×C22⋊A4 | C4 |
| Fitting, socle | C25×C3 | D3 |
| prime radical, cosocle | C2×C3×C22⋊A4 | C2 |
| LCS | G ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
334 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 29 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|a^2=b,c^a=c^2,d^a=d^2,e^a=f,f^a=e,g^a=h,h^a=g,e^c=f,f^c=ef,g^c=gh,h^c=g36 conjugacy classes — view the full character table.
| maximal subgroup of | #11920746, #11920747, #11920750, … |
| maximal normal subgroup of | #11920746, #11920747, #11920750, … |
| maximal quotient of | C3⋊#85561, C3⋊#85562, C3⋊#85563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8488);