Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of H(3,𝔽3)×AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2 |
| direct product | 3−1+2×#352 |
| unambiguous semidirect product | (C9×#352)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#99 |
| commutator | C23×C3 | C2×C32×C4 |
| Frattini, cosocle | C2×C3×D4 | C22×C32 |
| prime radical | C2×C32×D4 | C22×C3 |
| LCS | G ▷ C23×C3 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×3−1+2 ◁ C2×D4×3−1+2 ◁ G | |
272 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 91 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)(12,18)(13,21,22,16)(14,17)(15,19,23,25)(20,24);(2,5,6)(4,8,9)(10,12,14,20)(11,15,17,23)(13,19,22,25)(16,18,21,24)143 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2122);