Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #11720106 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C42⋊C27 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11720106 |
| commutator, hypocenter | C42⋊C27 | C4 |
| Frattini | C23×C9 | S4 |
| Fitting | C2×C42×C9 | D3 |
| socle, prime radical | C23×C3 | (C22⋊C9)⋊C2 |
| cosocle | C2×C42⋊C27 | C2 |
| LCS | G ▷ C42⋊C27 | |
| UCS | C1 ◁ C2 | |
126 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 21 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^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=fg,g^a=gi,h^a=hi,f^c=ghi,g^c=fgh,h^c=i,i^c=hi84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 191);