Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#20460 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C4 | C22×C42×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C22×C4⋊C4 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#117 ◁ G | |
994 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)(20,21,23,22)(24,25,28,34,30,33,29,35)(26,36,38,45,39,49,43,31)(27,32,41,46,40,48,42,47)(37,50,52,54,51,55,53,44);(24,26,29,38,30,39,28,43)(25,31,35,36,33,45,34,49)(27,37,42,52,40,51,41,53)(32,44,47,50,48,54,46,55);(20,22,23,21)(24,27,30,40)(25,32,33,48)(26,37,39,51)(28,41,29,42)(31,44,45,54)(34,46,35,47)(36,50,49,55)(38,52,43,53);(25,33)(27,40)(31,45)(34,35)(36,49)(37,51)(41,42)(52,53)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13448);