Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2 |
| direct product | 3−1+2×#349 |
| unambiguous semidirect product | (C9×#349)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C32×C22⋊C4 |
| commutator | C22×C3 | C2×C32×C8 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle | C2×C3 | C32×C22⋊C8 |
| prime radical | C23×C32 | C3×C8 |
| LCS | G ▷ C22×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C8×3−1+2 ◁ G | |
232 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 232 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) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,16,13,17,19,23,14,15,18,21,20,22,24,25);(2,5,6)(4,8,9)(11,15)(12,18)(17,22)(19,24)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2113);