Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17841 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C22×C4⋊C4 |
| prime radical | C2×#41×C19 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C4×#41×C19 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
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)(22,29)(23,27)(24,34)(25,28)(26,32)(30,31)(33,35)(36,37,40,44,41,45,39,43)(38,46,49,50,47,51,48,42);(20,22,23,31,25,33,24,32)(21,26,34,35,28,30,27,29)(36,38,40,49,41,47,39,48)(37,42,43,51,45,50,44,46);(20,21)(22,30)(23,34)(24,27)(25,28)(26,33)(29,31)(32,35)(38,47)(42,50)(46,51)(48,49);(20,23,25,24)(21,27,28,34)(22,31,33,32)(26,29,30,35)(36,39,41,40)(37,43,45,44)(38,48,47,49)(42,51,50,46)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10829);