Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12008 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C19 | D8 |
| commutator | C8 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×#107 |
| prime radical | C23×C19 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,24,23,25,26,27)(28,29)(30,39)(31,36)(32,46)(33,48)(34,38)(35,42)(37,40)(41,52)(43,58)(44,54)(45,51)(47,59)(49,56)(50,57)(53,55);(28,30,40,55,33,43,47,57,34,44,45,56,31,41,32,42)(29,35,46,52,36,49,51,54,38,50,59,58,48,53,37,39);(28,31,34,33)(29,36,38,48)(30,41,44,43)(32,45,47,40)(35,49,50,53)(37,46,51,59)(39,52,54,58)(42,56,57,55)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4996);