Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×#361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | #102 |
| commutator | C8 | C2×C33×C4 |
| Frattini, cosocle | C2×C8 | C22×C33 |
| prime radical | C2×C33×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C2×C33×C4 ◁ C2×C33×C8 ◁ G | |
868 conjugacy classes of subgroups; only the 420 normal subgroups are shown.
This group has 868 conjugacy classes of subgroups — too many to draw. Only its 420 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(10,11,13,17,14,19,24,18)(12,20,22,16,23,15,21,25);(4,5,6)(11,15)(12,21)(13,24)(16,17)(18,25)(19,20)(22,23);(7,8,9)432 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12217);