Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12614 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C19 | D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C8×D4 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C19 ◁ C22×C16×C19 ◁ G | |
426 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)(24,25,28,34,29,33,41,48,30,35,42,49,27,31,40,44)(26,32,37,46,38,45,51,54,39,47,52,55,36,43,50,53);(21,23)(25,31,35,33)(32,43,47,45)(34,44,49,48)(46,53,55,54);(24,26,28,37,29,38,41,51,30,39,42,52,27,36,40,50)(25,32,34,46,33,45,48,54,35,47,49,55,31,43,44,53)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5602);