Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#12561 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×#102 |
| prime radical | C24×C19 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ C23×C8×C19 ◁ G | |
570 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)(26,35)(27,40)(28,43)(29,45)(30,34)(31,37)(32,42)(33,41)(36,49)(38,46)(39,55)(44,48)(47,54)(50,53)(51,52);(24,26,28,38,41,52,42,53,30,36,27,37,29,39,44,54)(25,31,32,35,45,50,43,55,34,46,48,49,33,47,40,51);(26,36)(31,46)(35,49)(37,38)(39,52)(47,50)(51,55)(53,54)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5549);