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×#11176 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #394 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
834 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)(20,21)(22,23)(24,25)(26,36,39,31)(27,40,43,51)(28,34)(29,35)(30,33)(32,42,49,45)(37,48)(38,50,44,47)(41,52)(46,53)(54,55);(24,26)(25,31)(27,41)(28,38)(29,44)(30,39)(32,48)(33,36)(34,47)(35,50)(37,42)(40,46)(43,53)(45,54)(49,55)(51,52);(21,23)(24,27)(25,32,33,49)(26,37)(28,42)(29,45)(30,43)(31,46,36,52)(34,40,35,51)(38,53)(39,54)(41,44)(47,55,50,48)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4164);