Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22 |
| direct product | 3−1+2×#472 |
| unambiguous semidirect product | (C9×#472)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#121 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C32×#472 | C3 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C4×3−1+2 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(11,15)(12,19)(13,16)(18,23)(20,21)(22,24);(2,5,6)(4,8,9)(10,11,16,23,14,18,13,15)(12,17,19,24,21,25,20,22);(10,12)(11,17)(13,20)(14,21)(15,22)(16,19)(18,25)(23,24)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11239);