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×#21072 |
| 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 | C22×#41×C19 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×M16×C19 ◁ G | |
1018 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)(21,22)(24,25,29,34,30,33,28,42)(26,31,38,45,36,46,37,52)(27,39,40,32,41,48,43,47)(35,49,51,44,50,54,53,55);(20,21,23,22)(24,26,29,38,30,36,28,37)(25,31,34,45,33,46,42,52)(27,35,40,51,41,50,43,53)(32,44,48,54,47,55,39,49);(20,22,23,21)(24,27)(25,32)(26,35)(28,43)(29,40)(30,41)(31,44)(33,47)(34,48)(36,50)(37,53)(38,51)(39,42)(45,54)(46,55)(49,52);(25,33)(26,36)(32,47)(34,42)(35,50)(37,38)(39,48)(51,53)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14060);