Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11150 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #397 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25)(26,35)(27,39,43,53)(28,44)(29,33)(30,34)(31,37)(32,42,50,41)(36,46,40,49)(38,47)(45,48)(51,52,55,54);(20,21)(22,23)(24,26)(25,31)(27,40)(28,45)(29,37)(30,38)(32,49)(33,47)(34,48)(35,44)(36,43)(39,51)(41,52)(42,54)(46,50)(53,55);(21,23)(24,27,28,41,30,43,29,42)(25,32,33,39,34,50,44,53)(26,36,37,52,38,40,45,54)(31,46,35,51,48,49,47,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4138);