Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C22×D8×C19 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
480 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)(21,23)(24,25,30,37,33,42,29,41,32,38,28,35,31,26,39,36)(27,43,46,34,50,52,45,40,48,55,44,54,47,51,49,53);(20,21,22,23)(24,26,32,42)(25,33,38,31)(27,40,48,53)(28,37,30,35)(29,41,39,36)(34,47,54,50)(43,49,55,45)(44,51,46,52);(24,27,32,48)(25,34,38,54)(26,40,42,53)(28,45,30,49)(29,44,39,46)(31,50,33,47)(35,55,37,43)(36,52,41,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6006);