Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C19 · Schur multiplier (H2) = C27×C42 |
| direct product | C2×C4×C19×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42×C19 | C22 |
| commutator | C2 | C23×C42×C19 |
| Frattini, cosocle | C23 | C25×C19 |
| socle | C24×C19 | C24 |
| prime radical | C25×C19 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42×C19 ◁ G | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(3,4,5,6);(7,8,9,10)(11,12)(13,16)(14,15)(17,18);(12,15)(16,18);(11,13,14,17)(12,16,15,18)3040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26978);