Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#11038 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #394 |
| prime radical | C2×C19×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
1278 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)(26,36)(27,41,44,32)(28,35)(29,33)(30,34)(31,38)(37,49,52,54)(39,47)(40,48)(42,46,53,55)(43,50,45,51);(21,23)(24,26)(25,31)(27,42)(28,40)(29,38)(30,39)(32,49)(33,47)(34,48)(35,36)(37,43)(41,54)(44,53)(45,52)(46,50)(51,55);(24,27)(25,32)(26,37)(28,43)(29,45)(30,44)(31,46)(33,50)(34,41)(35,51)(36,49)(38,42)(39,52)(40,53)(47,54)(48,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4026);