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) = C22×C32×C4 · Schur multiplier (H2) = C23×C32 |
| direct product | H(3,𝔽3)×#421 |
| unambiguous semidirect product | (C32×#421)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×C32×D4 |
| commutator | C3×C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C2×C3 | C2×C32×C22⋊C4 |
| prime radical | C2×#41×H(3,𝔽3) | C2 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C4×H(3,𝔽3) ◁ G | |
847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,14,17,15,18,13,22)(12,19,21,16,20,25,23,24);(2,5,6)(4,8,9)(10,12)(11,16)(13,23)(14,21)(15,20)(17,25)(18,24)(19,22);(12,20)(16,24)(19,25)(21,23)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11085);