Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C210 |
| direct product | C23×D4×3−1+2 |
| unambiguous semidirect product | C4⋊(C24×3−1+2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C25×C32 |
| prime radical | C23×C32×D4 | C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
5448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(11,12,14,13,16,18,17,15,19);(3,4)(12,15,16)(14,18,19);(5,6);(7,8,9,10);(8,10)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31079);