Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17840 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C22×C4⋊C4 |
| prime radical | C2×#41×C19 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C4×#41×C19 ◁ G | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,25)(23,26)(27,32)(28,33)(29,35)(31,34)(36,37,40,43,41,44,39,49)(38,45,48,50,46,51,47,42);(20,21,26,33,24,28,23,25)(22,29,34,32,30,27,31,35)(36,38,40,48,41,46,39,47)(37,42,49,51,44,50,43,45);(20,22)(21,27)(23,31)(24,30)(25,32)(26,34)(28,29)(33,35)(38,46)(42,50)(45,51)(47,48);(22,30)(27,29)(31,34)(32,35)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10828);