Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C59 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C59×#1461 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C59 | C22⋊C4 |
| commutator | C22 | C23×C8×C59 |
| Frattini, cosocle | C22×C4 | C24×C59 |
| socle | C23×C59 | C2×C22⋊C4 |
| prime radical | C24×C59 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C59 ◁ C22×C42×C59 ◁ G | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(3,4,6,9,7,10,5,8)(11,12)(13,19,21,16)(14,17)(15,18)(20,25,26,23)(22,24);(11,13,15,21)(12,16,18,19)(14,20,22,26)(17,23,24,25);(3,5,7,6)(4,8,10,9)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)5192 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 10336);