Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#23017 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C19×#3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×C4⋊C8 ◁ G | |
864 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)(20,21,23,22)(24,25,29,34,30,33,28,42)(26,31,37,46,38,45,36,52)(27,39,40,32,41,48,43,47)(35,49,50,44,51,55,53,54);(21,22)(24,26,29,37,30,38,28,36)(25,31,34,46,33,45,42,52)(27,35,40,50,41,51,43,53)(32,44,48,55,47,54,39,49);(24,27)(25,32)(26,35)(28,43)(29,40)(30,41)(31,44)(33,47)(34,48)(36,53)(37,50)(38,51)(39,42)(45,54)(46,55)(49,52);(20,22,23,21)(25,33)(31,45)(32,47)(34,42)(39,48)(44,54)(46,52)(49,55)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16005);