Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 15925248 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C25×C3×C4 |
| abelian | C22×C3×C42×C9 |
| multiplicative | (ℤ/3705)ˣ, (ℤ/3848)ˣ, (ℤ/3885)ˣ, (ℤ/3952)ˣ, (ℤ/4144)ˣ, (ℤ/4940)ˣ, (ℤ/5180)ˣ, (ℤ/5328)ˣ, (ℤ/5616)ˣ, (ℤ/5772)ˣ, (ℤ/6660)ˣ, (ℤ/7020)ˣ, (ℤ/7410)ˣ, (ℤ/7770)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22×C3 | C24×C32 |
| socle, prime radical | C24×C32 | C22×C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
2490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21);(5,6,7,8)(22,23,24);(9,10);(11,12)1728 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20578);