Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C59 · Schur multiplier (H2) = C25 |
| direct product | C59×#33617 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C59 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C59 |
| socle | C2×C59 | C2×#466 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C59 ◁ C22×C4×C59 ◁ C2×#131×C59 ◁ G | |
642 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)(3,15)(4,22)(5,11)(6,28)(7,13)(8,14)(9,19)(10,25)(12,16)(17,45)(18,40)(20,41)(21,42)(23,35)(24,55)(26,38)(27,37)(29,36)(30,57)(31,50)(32,43)(33,51)(34,56)(39,48)(44,53)(46,61)(47,52)(49,62)(54,60)(58,63)(59,64)(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);(1,3,16,43,7,20,29,51,8,21,27,50,5,18,6,19)(2,9,28,40,11,31,37,42,14,33,36,41,13,32,12,15)(4,17,44,57,25,48,56,64,26,49,54,63,23,46,24,47)(10,30,53,45,22,52,55,61,35,58,60,62,38,59,34,39);(1,4)(2,10)(3,17)(5,23)(6,24)(7,25)(8,26)(9,30)(11,22)(12,34)(13,38)(14,35)(15,39)(16,44)(18,46)(19,47)(20,48)(21,49)(27,54)(28,53)(29,56)(31,52)(32,59)(33,58)(36,60)(37,55)(40,45)(41,62)(42,61)(43,57)(50,63)(51,64);(1,5,8,7)(2,11,14,13)(3,18,21,20)(4,23,26,25)(6,27,29,16)(9,31,33,32)(10,22,35,38)(12,28,37,36)(15,40,42,41)(17,46,49,48)(19,50,51,43)(24,54,56,44)(30,52,58,59)(34,53,55,60)(39,45,61,62)(47,63,64,57)3068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 26605);