Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#12657 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C42 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×#97 |
| prime radical | C23×C19 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ G | |
334 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)(26,36,39,51)(27,32)(28,33)(29,44)(30,35)(31,38,47,52)(34,45)(37,48,40,54)(41,55,53,46)(42,49)(43,50);(21,23)(24,26,29,40,28,38,42,53,30,39,45,37,43,52,27,41)(25,31,34,46,33,36,32,48,35,47,44,55,50,51,49,54);(24,27,28,29,30,42,43,45)(25,32,33,44,35,49,50,34)(26,37,52,53,39,40,38,41)(31,46,51,54,47,55,36,48)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5645);