Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#22690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | D42 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ G | |
846 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)(26,38)(27,42)(28,34)(29,36)(30,35)(31,37)(32,40)(33,43)(39,53)(41,51)(44,52)(45,55)(46,49)(47,50)(48,54);(24,26,31,41)(25,32,37,50)(27,28,44,46)(29,47,30,40)(33,49,53,34)(35,51,36,38)(39,48,43,45)(42,55,52,54);(24,27,30,43,31,44,29,39)(25,33,36,42,37,53,35,52)(26,28,40,45,41,46,47,48)(32,49,38,55,50,34,51,54);(24,28)(25,34)(26,39)(27,40)(29,48)(30,45)(31,46)(32,42)(33,51)(35,55)(36,54)(37,49)(38,53)(41,43)(44,47)(50,52)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 15678);