Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#21707 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C19×#3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#97 ◁ G | |
998 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)(21,23)(24,25,29,33,30,34,28,43)(26,35,38,45,39,49,37,31)(27,32,42,47,40,46,41,53)(36,48,52,54,50,55,51,44);(24,26,28,37,30,39,29,38)(25,31,33,35,34,45,43,49)(27,36,41,51,40,50,42,52)(32,44,47,48,46,54,53,55);(20,21,22,23)(27,40)(32,46)(36,50)(41,42)(44,54)(47,53)(48,55)(51,52);(24,27,29,42,30,40,28,41)(25,32,33,47,34,46,43,53)(26,36,38,52,39,50,37,51)(31,44,35,48,45,54,49,55)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14695);