Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#20768 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C22×C4⋊C4 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#117 ◁ G | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 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)(22,25)(23,24)(26,27)(28,29,33,37,34,38,32,48)(30,39,43,50,41,53,42,35)(31,44,46,51,47,58,45,36)(40,54,57,59,55,49,56,52);(21,24)(25,27)(28,30,32,42,34,41,33,43)(29,35,37,39,38,50,48,53)(31,40,45,56,47,55,46,57)(36,49,44,52,51,54,58,59);(20,22,23,26)(21,25,24,27)(28,31,32,45,34,47,33,46)(29,36,37,44,38,51,48,58)(30,40,42,56,41,55,43,57)(35,49,39,52,50,54,53,59);(30,41)(35,50)(39,53)(40,55)(42,43)(49,54)(52,59)(56,57)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13756);