Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#17718 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C22×C22⋊C4 |
| prime radical | C24×C19 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C42×C19 ◁ G | |
972 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)(20,21,24,29,25,27,23,28)(22,30,31,35,32,26,33,34)(36,37)(38,44)(39,47)(40,42)(41,45)(43,49)(46,51)(48,50);(20,22,25,32)(21,26,27,30)(23,31,24,33)(28,34,29,35)(36,38,40,46)(37,41,42,50)(39,48,49,45)(43,51,47,44);(21,27)(26,30)(28,29)(34,35)(37,42)(41,50)(43,47)(44,51);(20,23,25,24)(21,28,27,29)(22,31,32,33)(26,34,30,35)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10706);