Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C59 · Schur multiplier (H2) = C25 |
| direct product | C2×C59×#2168 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C59 | C23 |
| commutator | C22 | C23×C8×C59 |
| Frattini, cosocle | C2×C8 | C24×C59 |
| socle | C23×C59 | C22×C8 |
| prime radical | C22×D4×C59 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C59 ◁ G | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2)(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);(3,4)(5,6)(7,8,10,14,11,9,16,17,12,15,19,22,20,18,13,21);(4,6);(7,9,10,17,11,15,16,22,12,18,19,21,20,8,13,14)6608 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 26418);