Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#21003 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×#121 |
| prime radical | C2×C19×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
1686 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)(20,21)(22,23)(24,25,29,35,30,34,28,33)(26,31,37,45,38,47,39,46)(27,40,41,32,42,48,43,49)(36,50,52,44,53,54,51,55);(24,26,29,39,30,38,28,37)(25,31,33,45,34,47,35,46)(27,36,43,52,42,53,41,51)(32,44,48,50,49,55,40,54);(21,23)(24,27)(25,32)(26,36)(28,41)(29,43)(30,42)(31,44)(33,48)(34,49)(35,40)(37,51)(38,53)(39,52)(45,50)(46,54)(47,55);(24,28,30,29)(25,33,34,35)(26,37,38,39)(27,41,42,43)(31,45,47,46)(32,48,49,40)(36,51,53,52)(44,50,55,54)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13991);