Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×#356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | #99 |
| commutator | C2×C4 | C2×C33×C4 |
| Frattini, cosocle | C2×Q8 | C22×C33 |
| prime radical | C22×C33 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C22×C33 ◁ C2×C33×Q8 ◁ G | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(10,11,14,18,15,19,12,20)(13,17,21,24,23,25,22,16);(4,5,6)(10,12,15,14)(11,16,19,24)(13,22,23,21)(17,18,25,20);(7,8,9)351 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12212);