Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C59 · Schur multiplier (H2) = C26 |
| direct product | C2×C4×D16×C59 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C59 | D8 |
| commutator | C8 | C23×C4×C59 |
| Frattini, cosocle | C2×C8 | C24×C59 |
| socle | C23×C59 | C2×D8 |
| prime radical | C22×D16×C59 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C59 ◁ C2×C42×C59 ◁ C2×C4×C8×C59 ◁ G | |
806 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)(9,12)(10,19)(11,15)(13,16)(14,17)(18,21)(20,22);(8,12)(9,16)(10,17)(13,14)(15,21)(18,22)(19,20)5192 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 26432);