Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12608 |
| 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 | C24×C19 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#97 ◁ G | |
462 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)(20,21)(22,23)(24,25)(26,36,39,47)(27,33)(28,34)(29,35)(30,32)(31,38,44,37)(40,50,53,55)(41,48)(42,46)(43,49)(45,52,54,51);(21,23)(24,26,28,40,29,38,43,52,30,39,42,53,27,37,41,51)(25,31,34,45,35,36,49,50,32,44,46,54,33,47,48,55);(25,32)(26,37,39,38)(28,42)(31,36,44,47)(33,35)(40,52,53,51)(41,43)(45,55,54,50)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5596);