Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16×C19 · Schur multiplier (H2) = C2×C8 |
| direct product | C19×#11410 |
| subgroup | G/H | |
|---|---|---|
| center | C82×C19 | C22 |
| commutator | C2 | C2×C4×C16×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×C4×C8 |
| prime radical | C23×C19 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C82×C19 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35);(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35)(36,37,38,41,39,42,44,47,40,43,45,48,46,49,50,51);(22,28)(25,31)(29,34)(32,35)3040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4398);