Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#11498 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C23×C19 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,28,30,25)(23,26)(24,27)(29,34,35,32)(31,33)(36,37,41,47,43,46,56,64,42,48,55,65,40,57,39,45)(38,49,52,63,51,44,50,60,53,62,66,59,58,61,54,67);(20,22,24,30)(21,25,27,28)(23,29,31,35)(26,32,33,34)(36,38,42,53)(37,44,48,61)(39,54,56,50)(40,58,43,51)(41,52,55,66)(45,63,64,59)(46,62,57,49)(47,60,65,67);(20,23,24,31)(21,26,27,33)(22,29,30,35)(25,32,28,34)(36,39,40,55,42,56,43,41)(37,45,57,65,48,64,46,47)(38,50,58,52,53,54,51,66)(44,59,49,60,61,63,62,67)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4486);