Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#20355 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C19 |
| Frattini, cosocle | C24 | C24×C19 |
| prime radical | C24×C19 | C24 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×Q8×C19 ◁ G | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 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)(26,34,37,30)(27,31)(28,32)(29,33)(35,48,51,42)(36,49,53,43)(38,44)(39,45)(40,46)(41,47)(50,55,52,54);(20,21,22,23)(24,26)(25,30)(27,35)(28,36)(29,37)(31,42)(32,43)(33,34)(38,50)(39,51)(40,52)(41,53)(44,54)(45,48)(46,55)(47,49);(24,27,29,39)(25,31,33,45)(26,35,37,51)(28,40,41,38)(30,42,34,48)(32,46,47,44)(36,52,53,50)(43,55,49,54);(24,28,29,41)(25,32,33,47)(26,36,37,53)(27,38,39,40)(30,43,34,49)(31,44,45,46)(35,50,51,52)(42,54,48,55)1558 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13343);