Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#20446 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C4 | C22×C42×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×M16×C19 ◁ G | |
1010 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)(26,34,37,47)(27,31)(28,33)(29,32)(30,36,43,40)(35,48,49,54)(38,45)(39,44)(41,46)(42,50,53,52)(51,55);(20,21,22,23)(24,26)(25,30)(27,35)(28,40)(29,37)(31,42)(32,43)(33,47)(34,46)(36,41)(38,49)(39,52)(44,54)(45,53)(48,55)(50,51);(24,27,28,39,29,38,41,51)(25,31,33,44,32,45,46,55)(26,35,40,52,37,49,36,50)(30,42,47,54,43,53,34,48);(25,32)(27,38)(30,43)(33,46)(34,47)(35,49)(39,51)(50,52)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13434);