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) = C22×C32 |
| direct product | H(3,𝔽3)×#428 |
| unambiguous semidirect product | (C32×#428)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C32×D4 |
| commutator | C3×C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C23×H(3,𝔽3) | D4 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C4×H(3,𝔽3) ◁ G | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,12,13)(15,18)(16,20)(19,21);(2,5,6)(4,8,9)(14,15,16,18,17,19,20,21);(15,19)(18,21)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11106);