Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11194 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #386 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
822 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)(26,36)(27,32,43,51)(28,35)(29,33)(30,34)(31,38)(37,48,54,52)(39,46)(40,47)(41,45,53,55)(42,50,44,49);(20,21)(22,23)(24,26)(25,31)(27,41)(28,40)(29,38)(30,39)(32,48)(33,46)(34,47)(35,36)(37,44)(42,54)(43,53)(45,50)(49,55)(51,52);(21,23)(24,27,28,42,30,43,29,44)(25,32,33,49,34,51,35,50)(26,37,38,53,39,54,40,41)(31,45,36,52,47,55,46,48)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4182);