Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C59 · Schur multiplier (H2) = C25 |
| direct product | C59×#33441 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C59 | C2×D4 |
| commutator | C4 | C23×C8×C59 |
| Frattini, cosocle | C2×C8 | C24×C59 |
| socle | C2×C59 | C2×C8×D4 |
| prime radical | C59×#575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C59 ◁ C22×C16×C59 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2,6,13,7,12,26,37,8,11,27,36,5,23,25,49)(3,14,17,39,16,9,28,31,18,32,43,56,24,30,50,55)(4,10,20,35,21,34,47,58,22,33,48,57,19,44,46,61)(15,38,41,59,40,29,51,53,42,54,60,64,45,52,62,63)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3)(2,9)(4,15)(5,24)(6,17)(7,16)(8,18)(10,29)(11,30)(12,32)(13,31)(14,23)(19,45)(20,41)(21,40)(22,42)(25,50)(26,28)(27,43)(33,52)(34,54)(35,53)(36,55)(37,56)(38,44)(39,49)(46,62)(47,51)(48,60)(57,63)(58,64)(59,61);(1,4,6,20,7,21,26,47,8,22,27,48,5,19,25,46)(2,10,13,35,12,34,37,58,11,33,36,57,23,44,49,61)(3,15,17,41,16,40,28,51,18,42,43,60,24,45,50,62)(9,29,31,53,32,54,56,64,30,52,55,63,14,38,39,59);(2,11)(9,30)(10,33)(12,23)(13,36)(14,32)(29,52)(31,55)(34,44)(35,57)(37,49)(38,54)(39,56)(53,63)(58,61)(59,64)5192 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 26429);