Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(A4⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C22×C33 |
| unambiguous semidirect product | C3⋊(A4⋊SL(2,ℤ/4)) |
| unambiguous semidirect product | A4⋊(C3⋊SL(2,ℤ/4)) |
| derived | C3×A42 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(A4⋊S4) |
| commutator, hypocenter | C3×A42 | C4 |
| Fitting, socle | C25×C3 | C3⋊D3 |
| prime radical, cosocle | C2×C3×A42 | C2 |
| LCS | G ▷ C3×A42 | |
| UCS | C1 ◁ C2 | |
642 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 642 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3^3,2^4|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,f^a=fg,h^a=hi,h^c=i,i^c=hi,f^d=g,g^d=fg60 conjugacy classes — view the full character table.
| maximal subgroup of | #72242697, C4×C3⋊(A4⋊S4) |
| maximal normal subgroup of | #72242697, C4×C3⋊(A4⋊S4) |
| maximal quotient of | #72242699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46405);