Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#20510 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C4 | C22×C42×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,29,34,30,32,28,42)(26,35,37,31,38,46,43,44)(27,33,40,48,41,47,39,52)(36,49,50,45,51,55,53,54);(20,21,23,22)(24,26)(25,31)(27,36)(28,43)(29,37)(30,38)(32,44)(33,45)(34,46)(35,42)(39,53)(40,50)(41,51)(47,54)(48,55)(49,52);(20,22,23,21)(25,32)(31,44)(33,47)(34,42)(35,46)(45,54)(48,52)(49,55);(24,27,29,40,30,41,28,39)(25,33,34,48,32,47,42,52)(26,36,37,50,38,51,43,53)(31,45,46,55,44,54,35,49)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13498);