Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | GL(2,𝔽3)⋊(C22⋊C9) |
| unambiguous semidirect product | #5924⋊C9 |
| derived | #994 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | A4×S4 |
| commutator, hypocenter | #994 | C2×C9 |
| Fitting | C3×H(5,𝔽2) | C3×D3 |
| prime radical | C3×#5924 | C3 |
| cosocle | C3×#994 | C2×C3 |
| LCS | G ▷ #994 | |
| UCS | C1 ◁ C2×C3 | |
173 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3^2,2^2,3,2^3|b^3=c,d^2=i,e^2=i,g^2=i,h^2=i,f^a=f^2,g^a=gh,h^a=hi,d^b=de,e^b=d,e^d=ei,g^f=h,h^f=ghi,h^g=hi87 conjugacy classes — view the full character table.
| maximal subgroup of | #72253010 |
| maximal normal subgroup of | #72253010 |
| maximal quotient of | #72252981, #72252993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30109);