Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C42 · Schur multiplier (H2) = C2×C4 |
| direct product | 3−1+2×#405 |
| unambiguous semidirect product | (C9×#405)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C22×C32 |
| commutator | C2×C3 | C2×C32×C42 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C22×C3 | C22×C32×C4 |
| prime radical | C23×C32 | C2×C3×C4 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,4,3,6,8,7,5,9)(14,15,17,20,18,21,24,27)(16,19,23,26,22,25,28,29);(2,5,6)(4,8,9)(10,11,12,13)(14,16)(15,19)(17,23)(18,22)(20,26)(21,25)(24,28)(27,29);(16,22)(19,25)(23,28)(26,29)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11038);