Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3×C9×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C9 | D4 |
| commutator | C4 | C22×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C22×C32 | C3×C4⋊C4 |
| prime radical | C23×C32 | C3×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ C22×C3×C4×C9 ◁ G | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2)(19,20,21);(3,4,7,10,8,11,6,16)(5,12,14,17,15,18,13,9)(22,23,24,25,26,27,28,29,30);(3,5,6,13,8,15,7,14)(4,9,10,12,11,17,16,18)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11108);