Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | 3−1+2×#410 |
| unambiguous semidirect product | (C9×#410)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C32×C22⋊C4 |
| commutator | C22×C3 | C22×C32×C4 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle | C2×C3 | C2×C32×C22⋊C4 |
| prime radical | C2×C32×#41 | C2×C3 |
| LCS | G ▷ C22×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C4×3−1+2 ◁ G | |
632 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)(10,11)(12,18,20,15)(13,16)(14,17)(19,24,25,22)(21,23);(2,5,6)(4,8,9)(10,12)(11,15)(13,19)(14,20)(16,22)(17,18)(21,25)(23,24);(10,13,14,21)(11,16,17,23)(12,19,20,25)(15,22,18,24)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11053);