Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C9×#485 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C2×D4 |
| commutator | C2×C4 | C23×C3×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C22×C32 | C2×C3×D4 |
| prime radical | C23×C32 | C3×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C22×C3×C4×C9 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3)(13,14)(15,18)(16,17)(19,20)(21,22,25,29)(23,26,31,35)(24,27,32,28)(30,33,36,34);(4,5,6,7,8,9,10,11,12)(14,17)(18,20)(21,23,25,31)(22,26,29,35)(24,30,32,36)(27,33,28,34);(13,15,16,19)(14,18,17,20)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35)513 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11276);