Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#20678 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×#121 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
1330 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)(20,21)(22,23)(24,25,29,34,30,35,28,33)(26,36,39,47,40,46,38,31)(27,41,42,32,44,48,43,49)(37,51,52,50,54,45,53,55);(24,26,28,38,30,40,29,39)(25,31,34,36,35,47,33,46)(27,37,42,52,44,54,43,53)(32,45,41,50,49,51,48,55);(21,23)(24,27)(25,32)(26,37)(28,42)(29,43)(30,44)(31,45)(33,48)(34,41)(35,49)(36,50)(38,52)(39,53)(40,54)(46,55)(47,51);(24,28,30,29)(25,33,35,34)(26,38,40,39)(27,42,44,43)(31,46,47,36)(32,48,49,41)(37,52,54,53)(45,55,51,50)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13666);