Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #13134131 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived | C42⋊C27 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C42⋊C27 | C4 |
| Frattini | C23×C9 | S4 |
| Fitting | C9×#127 | D3 |
| socle | C22×C3 | (C22⋊C9)⋊C4 |
| prime radical | C3×#127 | D9 |
| cosocle | #127⋊C27 | C2 |
| LCS | G ▷ C42⋊C27 | |
| UCS | C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^2,3^3,2^4|a^2=b,c^3=d,d^3=e,f^2=h,g^2=i,c^a=c^2d^2e^2,d^a=d^2e^2,e^a=e^2,f^a=fgh,g^a=gh,h^a=hi,f^b=fh,g^b=gi,f^c=ghi,g^c=fgh,h^c=i,i^c=hi51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 194);