Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C59 · Schur multiplier (H2) = C28 |
| direct product | C59×#60322 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C59 | C25 |
| commutator | C22 | C24×C4×C59 |
| Frattini, cosocle | C2×C4 | C25×C59 |
| socle | C22×C59 | C24×C4 |
| prime radical | C59×#3459 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C59 ◁ G | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 5 generators on 83 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(1,3)(2,5)(4,8)(6,7)(11,17)(15,20)(18,23)(22,24);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23);(2,6)(5,7);(9,11,12,18,13,17,19,23)(10,15,16,22,14,20,21,24)4484 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 53310);