Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#17366 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C22×C22⋊C4 |
| prime radical | C22×#41×C19 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C42×C19 ◁ G | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1216 |
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)(20,21,23,27,24,25,22,26)(28,29)(30,38,41,34)(31,35)(32,36)(33,37)(39,43,40,42);(28,30)(29,34)(31,39)(32,40)(33,41)(35,42)(36,43)(37,38);(21,25)(26,27)(28,31,33,32)(29,35,37,36)(30,39,41,40)(34,42,38,43);(20,22,24,23)(21,26,25,27)(28,32,33,31)(29,36,37,35)(30,40,41,39)(34,43,38,42)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10354);