Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C59 · Schur multiplier (H2) = C23 |
| direct product | C59×#8880 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C59 | #120 |
| commutator | C22×C4 | C22×C4×C59 |
| Frattini, cosocle | C2×C42 | C23×C59 |
| socle | C22×C59 | #385 |
| prime radical | C59×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C59 ◁ C2×C42×C59 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2,5,8)(3,6,4,7)(9,10,13,19,15,20,14,18)(11,21,22,12,23,24,17,16)(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);(2,6)(3,4)(7,8)(9,11,14,17,15,23,13,22)(10,16,18,24,20,12,19,21);(2,7)(3,4)(6,8)(9,12)(10,17)(11,19)(13,24)(14,21)(15,16)(18,23)(20,22)2714 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 1868);