Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×3−1+2×C22⋊C8 |
| unambiguous semidirect product | (C2×C9×C22⋊C8)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22×C32 |
| commutator | C2×C3 | C22×C32×C8 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C23×C3 | C2×C32×C4 |
| prime radical | C24×C32 | C3×C4 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(15,16,18,17,20,22,21,19,23);(3,4)(5,6)(7,8,9,11,10,12,13,14)(16,19,20)(18,22,23);(4,6)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11041);