Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×#366 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C33 | D8 |
| commutator | C8 | C2×C33×C4 |
| Frattini, cosocle | C2×C8 | C22×C33 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ C2×C33×C4 ◁ C2×C33×C8 ◁ G | |
644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3)(10,11,12,13)(15,19)(16,20)(17,23)(21,29)(22,27)(24,28)(25,26);(4,5,6)(14,15,20,26,17,21,24,27,18,22,28,29,23,25,16,19);(7,8,9)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12222);