Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C25×C3 |
| direct product | C3×C9×#392 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C9 | C23 |
| commutator | C23 | C23×C3×C9 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle | C23×C32 | C23×C3 |
| prime radical | C32×#392 | C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C9 ◁ G | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3)(18,20)(21,22)(23,24);(4,5,6,7,8,9,10,11,12)(14,16)(18,20)(22,24);(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10997);