Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12850 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #117 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | #432 |
| prime radical | C24×C19 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#97 ◁ 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)(20,21)(22,23)(24,25,28,35,29,32,44,48,30,33,43,49,27,34,41,50)(26,36,39,51,40,45,53,54,37,31,42,46,38,47,52,55);(24,26)(25,31)(27,38)(28,42)(29,40)(30,37)(32,47)(33,36)(34,45)(35,51)(39,43)(41,53)(44,52)(46,49)(48,54)(50,55);(21,23)(25,32,33,34)(26,37)(28,43)(31,45,36,47)(35,50,49,48)(38,40)(41,44)(46,55,51,54)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5838);