Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×#534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C32 |
| direct product | H(3,𝔽3)×#536 |
| unambiguous semidirect product | (C32×#536)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22×C3 | C24×C32 |
| prime radical | C2×#41×H(3,𝔽3) | C2 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
1485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,4)(3,6,8)(5,9,7)(11,13)(15,19)(16,22)(17,25)(27,29);(2,5,6)(4,8,9)(14,15,18,19)(16,23,22,20)(17,21,25,26)(24,29,28,27);(10,11)(12,13)(14,16,18,22)(15,20,19,23)(17,24,25,28)(21,27,26,29);(14,17)(15,21)(16,24)(18,25)(19,26)(20,27)(22,28)(23,29)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20654);