Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C16 · Schur multiplier (H2) = C33 |
| direct product | C33×M64 |
| subgroup | G/H | |
|---|---|---|
| center | C33×C16 | C22 |
| commutator | C2 | C2×C33×C16 |
| Frattini, cosocle | C16 | C22×C33 |
| socle | C2×C33 | C2×C16 |
| prime radical | C22×C33 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C33×C16 ◁ G | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(4,5,6)(11,27)(13,29)(15,31)(17,33)(19,35)(21,37)(23,39)(25,41);(7,8,9)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12226);