Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C28 |
| direct product | C19×#20617 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#120 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C22×C19×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C4×C19 ◁ G | |
3610 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)(26,34)(27,31)(28,40)(29,32)(30,36)(33,43)(35,51)(37,48)(38,49)(39,46)(41,45)(42,47)(44,52)(50,53)(54,55);(24,26)(25,30)(27,35)(28,41)(29,37)(31,44)(32,45)(33,48)(34,40)(36,43)(38,54)(39,53)(42,52)(46,55)(47,50)(49,51);(24,27,29,39)(25,31,32,46)(26,35,37,53)(28,42,43,38)(30,44,45,55)(33,49,40,47)(34,50,48,51)(36,54,41,52);(25,32)(30,45)(31,46)(33,40)(34,48)(44,55)(47,49)(50,51)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13605);