Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C22×C32 |
| direct product | H(3,𝔽3)×#339 |
| unambiguous semidirect product | (C32×#339)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C32×D4 |
| commutator | C3×C4 | C32×C42 |
| Frattini, cosocle | C22×C3×C4 | C22×C32 |
| socle | C22×C3 | C32×C22⋊C4 |
| prime radical | C23×H(3,𝔽3) | D4 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C4×H(3,𝔽3) ◁ G | |
605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,12,13)(14,15)(16,18)(17,20)(19,21);(2,5,6)(4,8,9)(15,18,20,19)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2082);