Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13250 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #442 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
408 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)(26,36,39,52)(27,40,43,53)(28,44)(29,46)(30,35)(31,38,49,45)(32,42,50,54)(33,37)(34,47)(41,48)(51,55);(20,21,22,23)(24,26,37,54,47,43,28,45,30,39,55,42,29,27,41,38)(25,31,44,40,46,32,33,52,35,49,48,53,34,50,51,36);(24,27,41,54,29,39,55,38,30,43,28,42,47,26,37,45)(25,32,51,53,34,31,48,52,35,50,33,40,46,49,44,36)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6238);