Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12596 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #117 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | #432 |
| prime radical | C23×C19 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#97 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,29,34,28,36,44,51,30,35,46,32,45,33,27,42)(26,37,40,52,39,31,38,47,41,49,53,55,54,48,43,50);(20,21,22,23)(24,26)(25,31,35,48)(27,43)(28,39)(29,40)(30,41)(32,50,34,47)(33,37,36,49)(38,44)(42,52,51,55)(45,54)(46,53);(24,27,28,29,30,44,45,46)(25,32,33,51,35,34,36,42)(26,38,39,53,41,43,54,40)(31,47,37,52,48,50,49,55)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5584);