Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊#10596770 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊#10596770 |
| Fitting, socle | C24×C3 ⋊ #157 |
| derived | C3×A42 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C3×A42 | C4 |
| Fitting, socle | C24×C3 | #157 |
| prime radical, cosocle | C3×A4⋊S4 | C2 |
| LCS | G ▷ C3×A42 | |
| UCS | C1 | |
151 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^2,3^3,2^4|a^2=b,c^a=cd,d^a=cd^2,e^a=e^2,f^a=h,g^a=i,h^a=f,i^a=fg,c^b=c^2,d^b=d^2,g^b=fg,i^b=hi,f^c=g,g^c=fg,f^d=fg,g^d=f,h^d=i,i^d=hi27 conjugacy classes — view the full character table.
| maximal subgroup of | #72275395, #72275396, D3×#10596770, … |
| maximal normal subgroup of | #72275395, #72275396, D3×#10596770, … |
| maximal quotient of | C2×C3⋊#10596770 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47850);