Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#22896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | D42 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C2×C19×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ G | |
1016 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)(20,21,23,22)(24,25)(26,37)(27,33)(28,35)(29,34)(30,36)(31,40)(32,42)(38,46)(39,53)(41,52)(43,51)(44,48)(45,49)(47,55)(50,54);(20,22,23,21)(24,26)(25,31)(27,38)(28,44)(29,40)(30,41)(32,46)(33,50)(34,52)(35,37)(36,48)(39,42)(43,55)(45,54)(47,49)(51,53);(24,27,30,43)(25,32,36,49)(26,38,41,55)(28,42,29,45)(31,46,48,47)(33,35,51,34)(37,53,52,50)(39,40,54,44);(24,27)(25,33)(26,39)(28,45)(29,42)(30,43)(31,47)(32,34)(35,49)(36,51)(37,53)(38,44)(40,55)(41,54)(46,48)(50,52)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 15884);