Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#8781 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #117 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #382 |
| prime radical | C23×C19 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C19×#376 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,25,22,24,26,27)(28,29)(30,38,41,34)(31,42,44,57)(32,37)(33,36)(35,43,48,58)(39,52)(40,51,55,47)(45,49)(46,53)(50,59)(54,56);(21,24)(25,27)(28,30,32,40,33,41,45,55)(29,34,37,47,36,38,49,51)(31,39,43,53,44,54,58,59)(35,46,42,56,48,50,57,52);(28,31)(29,35,36,48)(30,39)(32,43)(33,44)(34,46,38,50)(37,42,49,57)(40,53)(41,54)(45,58)(47,56,51,52)(55,59)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1769);