Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#11616 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C19×#121 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,27,25,22,23,26)(28,29,31,36,32,37,41,49,33,35,42,47,43,48,53,57)(30,34,39,44,40,45,52,54,38,46,50,55,51,56,58,59);(20,22)(21,25)(23,27)(24,26)(28,30)(29,34)(31,39)(32,40)(33,38)(35,46)(36,44)(37,45)(41,52)(42,50)(43,51)(47,55)(48,56)(49,54)(53,58)(57,59);(20,23,25,24)(21,26,22,27)(29,35)(30,38)(36,47)(37,48)(39,50)(40,51)(49,57)(52,58)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4604);