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) = C22×C33 |
| direct product | C33×#354 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | #99 |
| commutator | C2×C4 | C2×C33×C4 |
| Frattini, cosocle | C2×D4 | C22×C33 |
| prime radical | C2×C33×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C22×C33 ◁ C2×C33×D4 ◁ G | |
980 conjugacy classes of subgroups; only the 364 normal subgroups are shown.
This group has 980 conjugacy classes of subgroups — too many to draw. Only its 364 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)(12,18,20,25)(13,21,23,16)(14,17)(15,19)(22,24);(4,5,6)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18);(7,8,9)351 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12210);