Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C19×D4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C19 | D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C23×C19 | C2×C22⋊C4 |
| prime radical | C22×D4×C19 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C19 ◁ C43×C19 ◁ G | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(5,6)(7,9)(8,11)(10,12)(13,14,15,17,16,18,19,20);(6,9)(7,12)(10,11)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1156);