Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#11018 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #386 |
| prime radical | C2×C19×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 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)(24,25,29,33,30,35,28,34)(26,36,38,31,39,46,44,47)(27,32,42,49,43,50,41,51)(37,48,52,45,53,55,40,54);(24,26)(25,31)(27,40)(28,44)(29,38)(30,39)(32,48)(33,46)(34,36)(35,47)(37,42)(41,53)(43,52)(45,49)(50,55)(51,54);(21,23)(24,27)(25,32)(26,37)(28,41)(29,42)(30,43)(31,45)(33,49)(34,51)(35,50)(36,48)(38,52)(39,53)(40,44)(46,55)(47,54)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4006);