Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#11001 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #379 |
| prime radical | C19×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
1306 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)(27,39,41,48)(28,42)(29,33)(30,34)(31,37)(32,40,49,44)(36,51,54,45)(38,46)(43,47)(50,53,55,52);(20,21)(22,23)(24,26)(25,31)(27,36)(28,43)(29,37)(30,38)(32,45)(33,46)(34,47)(35,42)(39,50)(40,53)(41,54)(44,52)(48,55)(49,51);(21,23)(24,27)(25,32)(26,36)(28,40)(29,44)(30,41)(31,45)(33,48)(34,49)(35,50)(37,52)(38,54)(39,42)(43,53)(46,55)(47,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 3989);