Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#33334 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C24 |
| commutator | C22 | C23×C8×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C23×C8 |
| prime radical | C22×#41×C19 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ G | |
1172 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)(24,25,27,31,28,32,37,43,29,33,38,44,39,45,49,53)(26,30,35,41,36,42,48,52,34,40,46,50,47,51,54,55);(21,23);(24,26)(25,30)(27,35)(28,36)(29,34)(31,41)(32,42)(33,40)(37,48)(38,46)(39,47)(43,52)(44,50)(45,51)(49,54)(53,55);(26,34)(30,40)(35,46)(36,47)(41,50)(42,51)(48,54)(52,55)1900 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26322);