Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C32 |
| direct product | H(3,𝔽3)×#575 |
| unambiguous semidirect product | (C32×#575)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×C32×D4 |
| commutator, Frattini, cosocle | C3×C4 | C24×C32 |
| socle | C2×C3 | C22×C32×D4 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C4×H(3,𝔽3) ◁ G | |
1441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,4)(3,6,8)(5,9,7)(10,11)(12,19)(13,18)(14,22)(15,17)(16,20)(21,25)(23,24);(2,5,6)(4,8,9)(10,12)(11,16)(13,20)(14,23)(15,21)(17,24)(18,25)(19,22);(11,17)(16,24)(18,22)(19,25);(10,13,15,14)(11,18,17,22)(12,20,21,23)(16,25,24,19)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20771);