Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C22⋊C4×3−1+2 |
| unambiguous semidirect product | (C4×C9×C22⋊C4)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22×C32 |
| commutator | C2×C3 | C2×C32×C42 |
| Frattini, socle, cosocle | C23×C3 | C23×C32 |
| prime radical | C24×C32 | C22×C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4)(13,14,16,15,18,20,19,17,21);(5,6,8,10)(7,9,11,12)(14,17,18)(16,20,21);(6,9)(10,12)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10954);