Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13726 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C19 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C19 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C42×C19 ◁ C4×C8×C19 ◁ C4×C16×C19 ◁ G | |
230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 4864 |
A minimal faithful permutation representation — 3 generators on 87 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,30,35)(26,36,41,62)(27,42,45,74)(28,46,48,76)(29,49,47,34)(31,50,53,38)(32,54,56,37)(33,57,58,43)(39,68,70,87)(40,71,69,61)(44,75,64,73)(51,81,82,86)(52,67,77,80)(55,65,72,83)(59,79,84,63)(60,78,85,66);(24,26,37,63,28,39,64,80,47,69,83,81,57,78,45,50,30,41,54,79,48,70,44,67,29,40,65,86,43,66,27,38)(25,31,42,60,33,51,72,71,49,77,75,87,76,84,56,62,35,53,74,85,58,82,55,61,34,52,73,68,46,59,32,36);(20,21,22,23)1330 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6714);