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) = C23×C32 |
| direct product | H(3,𝔽3)×#383 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23×C3 | C23×C32 |
| commutator | C22×C3 | C22×C32×C4 |
| prime radical | C23×H(3,𝔽3) | C23 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,12,13)(16,20)(18,21)(23,26)(24,28);(2,5,6)(4,8,9)(14,15,17,19)(16,18,20,21)(22,23)(24,27)(25,26)(28,29);(14,16)(15,18)(17,20)(19,21)(22,24,25,28)(23,27,26,29)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10971);