Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #459 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
294 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)(20,21,23,22)(24,25)(26,36)(27,33)(28,43)(29,46)(30,35)(31,37)(32,41)(34,45)(38,51)(39,53)(40,47)(42,52)(44,48)(49,54)(50,55);(20,22,23,21)(24,26)(25,31)(27,37)(28,44)(29,38)(30,40)(32,47)(33,51)(34,48)(35,50)(36,43)(39,45)(41,54)(42,55)(46,49)(52,53);(24,27,29,41,30,42,45,28)(25,32,34,33,35,43,46,52)(26,37,38,54,40,55,39,44)(31,47,48,51,50,36,49,53)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6168);