Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C26 |
| direct product | C61×#24296 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C61 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C61 ◁ C22×C4×C61 ◁ G | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2)(3,14)(4,19)(5,25)(6,29)(7,12)(8,13)(9,17)(10,23)(11,26)(15,31)(16,44)(18,33)(20,35)(21,36)(22,38)(24,37)(27,39)(28,53)(30,34)(32,46)(40,45)(41,59)(42,56)(43,57)(47,49)(48,63)(50,52)(51,60)(54,58)(55,61)(62,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3,6,17,8,18,7,30)(2,9,12,14,13,34,29,33)(4,20,22,15,24,42,23,43)(5,16,26,45,28,46,27,54)(10,35,37,57,38,56,19,31)(11,32,25,40,39,44,53,58)(21,48,49,41,51,61,50,62)(36,59,52,63,60,64,47,55);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,31)(11,36)(12,37)(13,38)(14,35)(16,41)(17,42)(18,43)(19,29)(20,30)(25,52)(26,49)(27,50)(28,51)(32,55)(33,56)(34,57)(39,60)(40,59)(44,63)(45,61)(46,62)(47,53)(48,54)(58,64);(1,5,7,27,8,28,6,26)(2,11,29,53,13,39,12,25)(3,16,30,54,18,46,17,45)(4,21,23,50,24,51,22,49)(9,32,33,58,34,44,14,40)(10,36,19,47,38,60,37,52)(15,41,20,48,43,62,42,61)(31,55,56,64,57,63,35,59)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 17284);