Finite Group DatabasebetaFinite Group Databasebeta| order | 1584 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 92160 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C22×C32×C4×C11 |
| multiplicative | (ℤ/2093)ˣ, (ℤ/2345)ˣ, (ℤ/2613)ˣ, (ℤ/2691)ˣ, (ℤ/3015)ˣ, (ℤ/3484)ˣ, (ℤ/4186)ˣ, (ℤ/4690)ˣ, (ℤ/5226)ˣ, (ℤ/5382)ˣ, (ℤ/6030)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C23×C32×C11 |
| socle, prime radical | C23×C32×C11 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1584 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4)(9,10,11)(15,16,17,18,19,20,21,22,23,24,25);(5,6)(12,13,14);(7,8)1584 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1584, 648);