Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | 3−1+2×#387 |
| unambiguous semidirect product | (C9×#387)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23×C3 | C23×C32 |
| commutator | C22×C3 | C22×C32×C4 |
| prime radical | C23×C32 | C23×C3 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(11,13)(14,15,17,16)(19,22)(20,24);(2,5,6)(4,8,9)(14,16,17,15)(18,19)(20,23)(21,22)(24,25);(10,11)(12,13)(18,20,21,24)(19,23,22,25)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10984);