Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#8817 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C19 | #127 |
| commutator | C42 | C22×C4×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C42×C19 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 4864 |
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)(26,32)(27,36)(28,30)(29,34)(31,33)(35,38)(37,39)(40,41)(42,48)(43,45)(44,52)(46,50)(47,51)(49,54)(53,55);(24,26,33,39,28,35,27,34)(25,29,36,38,30,37,31,32)(41,45)(46,53)(47,52)(48,54);(25,30)(29,37)(31,36)(32,38)(40,42,44,50,43,49,51,55)(41,46,47,48,45,53,52,54)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1805);