Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 48522240 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C36 |
| abelian | C34×C4 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C2×C34 |
| socle, prime radical | C2×C34 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4)(5,6,7);(8,9,10);(11,12,13);(14,15,16)324 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 159);