Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C19×#1384 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #106 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | #340 |
| prime radical | C24×C19 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ C23×C8×C19 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(3,4)(5,15,17,29)(6,12)(7,22)(8,24)(9,14)(10,16,25,30)(11,20)(13,23)(18,33,31,27)(19,34,32,26)(21,28);(3,5,7,18,23,32,21,30,9,17,20,31,8,19,6,16)(4,10,12,26,24,33,11,15,14,25,28,34,13,27,22,29)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5552);