Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C32 |
| direct product | H(3,𝔽3)×#407 |
| unambiguous semidirect product | (C32×#407)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| commutator, socle | C22×C3 | C22×C32×C4 |
| prime radical | C24×H(3,𝔽3) | C4 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
1045 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 1045 conjugacy classes of subgroups — too many to draw. Only its 315 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4)(3,6,8)(5,9,7)(10,11)(12,13)(14,15,16,19,17,18,20,21);(2,5,6)(4,8,9)(11,13);(15,18)(19,21)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11043);