Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C59 · Schur multiplier (H2) = C29 |
| direct product | C59×#60147 |
| 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 | D4×#41×C59 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C59 ◁ G | |
2742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 5 generators on 79 points:
(1,2)(3,4)(5,6,10,7,9,11,17,13)(8,12,16,14,15,18,20,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(2,4)(6,11)(7,13)(12,18)(14,19);(5,7,10,11,9,13,17,6)(8,14,16,18,15,19,20,12);(5,8)(6,12)(7,14)(9,15)(10,16)(11,18)(13,19)(17,20);(8,15)(12,18)(14,19)(16,20)5192 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 53135);