Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C4×C8 · Schur multiplier (H2) = C23×C33 |
| direct product | C33×#336 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C33×C4 | C22 |
| commutator | C2 | C33×C4×C8 |
| Frattini, cosocle | C22×C4 | C22×C33 |
| socle, prime radical | C23×C33 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C33×C4 ◁ G | |
1652 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)(18,19,20,22,21,23,24,25);(4,5,6)(11,13)(14,15,16,17);(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, 12192);