Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#21708 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C19×#3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,29,33,30,34,28,43)(26,35,39,44,36,48,38,31)(27,32,41,46,42,47,40,53)(37,49,52,54,50,55,51,45);(24,26,28,38,30,36,29,39)(25,31,33,35,34,44,43,48)(27,37,40,51,42,50,41,52)(32,45,46,49,47,54,53,55);(20,21,22,23)(26,36)(31,44)(35,48)(37,50)(38,39)(45,54)(49,55)(51,52);(24,27,29,41,30,42,28,40)(25,32,33,46,34,47,43,53)(26,37,39,52,36,50,38,51)(31,45,35,49,44,54,48,55)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14696);