Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 55296 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C22×C32×C4×C7 |
| multiplicative | (ℤ/1505)ˣ, (ℤ/1677)ˣ, (ℤ/1827)ˣ, (ℤ/1911)ˣ, (ℤ/1935)ˣ, (ℤ/2205)ˣ, (ℤ/2236)ˣ, (ℤ/2548)ˣ, (ℤ/3010)ˣ, (ℤ/3354)ˣ, (ℤ/3654)ˣ, (ℤ/3822)ˣ, (ℤ/3870)ˣ, (ℤ/4410)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C23×C32×C7 |
| socle, prime radical | C23×C32×C7 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 1008 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4)(9,10,11)(15,16,17,18,19,20,21);(5,6)(12,13,14);(7,8)1008 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 876);