Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#17316 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C19 | C24 |
| commutator | C2 | C22×C4×C8×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C22×C42 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C19 ◁ G | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(28,29,31,35,32,33,38,40)(30,34,37,41,36,39,42,43);(20,21,22,24,23,25,26,27)(29,33)(34,39)(35,40)(41,43);(28,30)(29,34)(31,37)(32,36)(33,39)(35,41)(38,42)(40,43);(30,36)(34,39)(37,42)(41,43)2584 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10304);