Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13162 |
| 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 | C23×C19 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C22×C16×C19 ◁ G | |
290 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)(22,23)(24,25,29,35,30,34,45,51,31,36,44,33,28,46,27,42)(26,32,39,47,40,48,37,52,41,49,53,50,38,54,43,55);(21,23)(24,26,29,39,30,40,45,37,31,41,44,53,28,38,27,43)(25,32,35,47,34,48,51,52,36,49,33,50,46,54,42,55);(24,27,28,44,31,45,30,29)(25,33,34,42,36,35,46,51)(26,37,38,39,41,43,40,53)(32,47,48,52,49,50,54,55)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6150);