Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33409 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C22×D8 |
| prime radical | C24×C19 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ C23×C8×C19 ◁ G | |
674 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)(26,35)(27,32)(28,42)(29,44)(30,34)(31,39)(33,36)(37,48)(38,49)(40,54)(41,47)(43,55)(45,52)(46,53)(50,51);(24,26,36,51,29,40,43,52,30,37,41,53,27,38,28,39)(25,31,42,49,32,46,47,48,34,45,55,54,44,50,33,35);(26,37)(31,45)(35,48)(38,40)(39,52)(46,50)(49,54)(51,53);(24,27,30,29)(25,32,34,44)(26,38,37,40)(28,41,43,36)(31,46,45,50)(33,42,47,55)(35,49,48,54)(39,53,52,51)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26397);