Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C42 · Schur multiplier (H2) = C23×C33 |
| direct product | C33×#340 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C33 | D4 |
| commutator | C4 | C33×C42 |
| Frattini, cosocle | C22×C4 | C22×C33 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C33 ◁ C22×C33×C4 ◁ G | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(10,11)(12,13)(14,15,16,17)(19,23)(20,21)(24,25);(4,5,6)(11,13)(18,19,21,24,22,25,20,23);(7,8,9)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12196);