Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#22465 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | D42 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C2×C19×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ G | |
1312 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)(24,25,29,35,31,27,30,34)(26,36,40,37,41,48,39,32)(28,33,45,50,46,42,44,51)(38,52,43,53,55,49,54,47);(24,26)(25,32)(27,37)(28,43)(29,40)(30,39)(31,41)(33,49)(34,48)(35,36)(38,44)(42,52)(45,55)(46,54)(47,50)(51,53);(21,23)(24,27,31,25)(26,37,41,32)(28,42,46,33)(29,35,30,34)(36,39,48,40)(38,53,55,47)(43,52,54,49)(44,51,45,50);(24,28)(25,33)(26,38)(27,42)(29,45)(30,44)(31,46)(32,47)(34,51)(35,50)(36,52)(37,53)(39,54)(40,43)(41,55)(48,49)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 15453);