Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17969 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C19×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C19×#431 ◁ G | |
582 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)(23,30)(26,32)(27,31)(29,34)(33,35)(36,37,41,44,42,38,40,46)(39,43,49,50,47,45,48,51);(20,21,24,27)(22,26,28,33)(23,31,30,25)(29,35,34,32)(36,38,42,37)(39,45,47,43)(40,44,41,46)(48,50,49,51);(20,22)(21,26)(23,29)(24,28)(25,32)(27,33)(30,34)(31,35)(36,39,41,49,42,47,40,48)(37,43,44,50,38,45,46,51);(22,28)(26,33)(29,34)(32,35)(37,38)(39,47)(44,46)(48,49)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10957);