Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42×C9 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C3×C4×C9×C4⋊C4 |
| unambiguous semidirect product | C4⋊(C3×C42×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4×C9 | C22 |
| commutator | C2 | C2×C3×C42×C9 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle, prime radical | C23×C32 | C23×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4×C9 ◁ G | |
970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3)(13,14,15,16);(4,5,6,7,8,9,10,11,12)(18,20)(21,22,23,24);(17,18,19,20)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10955);