Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of H(3,𝔽3)×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2 |
| direct product | 3−1+2×#367 |
| unambiguous semidirect product | (C9×#367)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C32×D8 |
| commutator | C3×C8 | C2×C32×C4 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| prime radical | C22×C32 | C3×D8 |
| LCS | G ▷ C3×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C4×3−1+2 ◁ C2×C8×3−1+2 ◁ G | |
184 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,13)(15,19)(16,23)(17,24)(20,25)(21,29)(22,28)(26,27);(2,5,6)(4,8,9)(14,15,20,29,17,21,16,19,18,22,23,27,24,26,25,28)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2167);