Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#17385 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C22×C22⋊C4 |
| prime radical | C24×C19 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C42×C19 ◁ G | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
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)(22,30,33,26)(23,27)(24,28)(25,29)(31,35,32,34)(36,37,40,44,41,45,39,43)(38,42,48,51,46,49,47,50);(20,22,25,33)(21,26,29,30)(23,31,24,32)(27,34,28,35)(36,38)(37,42)(39,47)(40,48)(41,46)(43,50)(44,51)(45,49);(20,23,25,24)(21,27,29,28)(22,31,33,32)(26,34,30,35)(38,46)(42,49)(47,48)(50,51);(20,24,25,23)(21,28,29,27)(22,32,33,31)(26,35,30,34)(36,39,41,40)(37,43,45,44)(38,47,46,48)(42,50,49,51)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10373);