Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C19×#10803 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | C4×C22⋊C4 |
| prime radical | C24×C19 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C8×C19 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,22)(24,25)(26,30)(27,37,38,31)(28,33)(29,32)(34,46,47,41)(35,43)(36,42)(39,50,51,44)(40,45)(48,54,55,52)(49,53);(20,21,23,22)(24,26,28,35,29,36,40,49)(25,30,33,43,32,42,45,53)(27,34,39,48,38,47,51,55)(31,41,44,52,37,46,50,54);(20,22,23,21)(24,27)(25,31)(26,34)(28,39)(29,38)(30,41)(32,37)(33,44)(35,48)(36,47)(40,51)(42,46)(43,52)(45,50)(49,55)(53,54)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 3791);