Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11186 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #394 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C2×C19 | #1440 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
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)(22,32,36,27)(23,37)(24,31)(25,30)(26,29)(28,39)(33,49,50,45)(34,43,35,44)(38,48,51,42)(40,47)(41,46);(20,22)(21,27)(23,38)(24,35)(25,34)(26,36)(28,45)(29,32)(30,44)(31,43)(33,39)(37,42)(40,51)(41,50)(46,49)(47,48);(20,23)(21,28,29,46)(22,33)(24,39)(25,41)(26,40)(27,42,32,48)(30,37,31,47)(34,38)(35,51)(36,50)(43,45,44,49)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4174);