Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of H(3,𝔽3)×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23 |
| direct product | C2×3−1+2×D16 |
| unambiguous semidirect product | (C2×C9×D16)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C32×D8 |
| commutator, Frattini, cosocle | C3×C8 | C23×C32 |
| prime radical | C2×C32×D16 | C3 |
| LCS | G ▷ C3×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C4×3−1+2 ◁ C2×C8×3−1+2 ◁ G | |
392 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(19,20,22,21,24,26,25,23,27);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(20,23,24)(22,26,27);(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11338);