Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#9236 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C42 | C22×C4×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #386 |
| prime radical | C23×C19 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,27,25,28,23,33)(22,29,32,34,30,35,31,26)(36,37)(38,42,39,47)(40,45)(41,44)(43,48,51,46)(49,50);(20,22)(21,26,28,34)(23,31)(24,32)(25,30)(27,29,33,35)(36,38)(37,42,44,47)(39,41)(40,48)(43,45,51,50)(46,49);(22,30)(26,34)(29,35)(31,32)(36,39,41,38)(37,43,44,51)(40,46,49,48)(42,50,47,45)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 2224);