Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 3919104 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C33 |
| abelian | C23×C32×C9 |
| multiplicative | (ℤ/1197)ˣ, (ℤ/2394)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C23×C33 |
| socle, prime radical | C23×C33 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(7,8,9,10,11,12,13,14,15);(3,4)(16,17,18);(5,6)(19,20,21)648 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 637);