Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C22×C42 |
| direct product | C19×#7854 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C19 | C42 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | C2×#95 |
| prime radical | C24×C19 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42×C19 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,26,24,27,22,25)(28,29)(30,36,38,42)(31,34)(32,35)(33,37,40,43)(39,41);(28,30)(29,33,35,40)(31,37)(32,38)(34,36,41,42)(39,43);(20,22,24,23)(21,25,27,26)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 842);