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) = C22×C32×C4 |
| direct product | C4×C4⋊C4×H(3,𝔽3) |
| unambiguous semidirect product | C4⋊(C42×H(3,𝔽3)) |
| unambiguous semidirect product | (C32×C4×C4⋊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 | C23×H(3,𝔽3) | C23 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
1067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4)(13,14,16)(15,18,20)(17,21,19);(6,8)(9,10,11,12)(14,17,18)(16,20,21);(5,6,7,8)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10956);