Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22 |
| direct product | 3−1+2×#468 |
| unambiguous semidirect product | (C9×#468)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C32×D4 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C32×#121 | C2×C3 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C4×3−1+2 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,3,6,8,7,5,9)(10,11)(12,13)(15,18)(16,20)(19,21);(2,5,6)(4,8,9)(14,15,16,18,17,19,20,21);(11,13)(15,19)(18,21)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11227);