Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C9×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8×C9 | D4 |
| commutator | C4 | C2×C3×C8×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle | C2×C32 | C3×C22⋊C8 |
| prime radical | C32×#41 | C3×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8×C9 ◁ C2×C3×C8×C9 ◁ G | |
280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3)(13,14,17,22,18,23,32,40,19,24,33,41,16,21,30,38)(15,25,27,42,26,36,43,39,28,37,31,44,29,20,34,35);(4,5,6,7,8,9,10,11,12)(13,15)(14,20)(16,29)(17,31)(18,26)(19,28)(21,37)(22,39)(23,25)(24,36)(27,33)(30,43)(32,34)(35,41)(38,42)(40,44)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2114);