Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#21014 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #386 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C24 | C24×C19 |
| socle | C2×C19 | C2×#386 |
| prime radical | C19×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 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)(22,26,32,39)(23,33,35,27)(24,28)(25,29)(30,37)(31,38,44,49)(34,46,47,40)(36,41)(42,48)(43,45)(50,51);(20,22)(21,26,29,39)(23,30)(24,31)(25,32)(27,37,33,45)(28,38,41,49)(34,42)(35,43)(36,44)(40,48,46,51)(47,50);(20,23)(21,27)(22,30)(24,34)(25,35)(26,37)(28,40)(29,33)(31,42)(32,43)(36,47)(38,48)(39,45)(41,46)(44,50)(49,51);(20,24,25,36)(21,28,29,41)(22,31,32,44)(23,34,35,47)(26,38,39,49)(27,40,33,46)(30,42,43,50)(37,48,45,51)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14002);