Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#33349 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C22×#99 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C2×D4 | C24×C19 |
| prime radical | C19×#3486 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×Q8×C19 ◁ C2×C19×2−1+4 ◁ G | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,27)(23,28)(24,36,37,26)(25,29)(30,40)(31,45,46,38)(32,41)(33,42)(34,48,49,39)(35,43)(44,51,47,50);(21,26)(27,38)(28,39)(29,36)(40,50)(41,45)(42,51)(43,48);(20,22,25,32)(21,27,29,41)(23,33,35,30)(24,31,37,46)(26,38,36,45)(28,42,43,40)(34,47,49,44)(39,51,48,50);(20,23,25,35)(21,28,29,43)(22,30,32,33)(24,34,37,49)(26,39,36,48)(27,40,41,42)(31,44,46,47)(38,50,45,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26337);