Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C19×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | #390 |
| prime radical | C22×C19×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 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)(5,9,14,23)(6,16)(7,19)(8,11)(10,17)(12,20)(13,25,26,34)(15,29,28,22)(18,30)(21,32,33,27)(24,31);(3,5)(4,9,11,23)(6,13)(7,15)(8,14)(10,21,24,33)(12,22,19,29)(16,25,30,34)(17,27)(18,26)(20,28)(31,32);(3,6,7,17,8,18,20,31)(4,10,12,16,11,24,19,30)(5,13,15,27,14,26,28,32)(9,21,22,25,23,33,29,34)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14231);