Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#21164 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C8×C19 ◁ G | |
890 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)(20,21)(22,23)(24,25,29,33,30,34,28,43)(26,31,37,45,38,46,36,52)(27,39,42,47,40,53,41,32)(35,48,51,54,49,55,50,44);(21,23)(24,26,29,37,30,38,28,36)(25,31,33,45,34,46,43,52)(27,35,42,51,40,49,41,50)(32,44,39,48,47,54,53,55);(24,27,28,41,30,40,29,42)(25,32,33,39,34,47,43,53)(26,35,36,50,38,49,37,51)(31,44,45,48,46,54,52,55);(27,40)(32,47)(35,49)(39,53)(41,42)(44,54)(48,55)(50,51)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14152);