Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13196 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C8×D4 |
| prime radical | C2×C19×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C22×C16×C19 ◁ G | |
430 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)(24,25,30,36,31,35,46,52,28,34,38,33,29,47,27,44)(26,37,42,51,43,53,39,54,40,32,48,49,41,50,45,55);(20,21,22,23)(24,26,38,48,31,43,27,45,28,40,30,42,29,41,46,39)(25,32,33,51,35,50,44,54,34,37,36,49,47,53,52,55);(24,27,29,38,28,46,31,30)(25,33,35,44,34,36,47,52)(26,39,41,42,40,45,43,48)(32,49,50,55,37,51,53,54)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6184);