Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#13311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C61 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C61 |
| prime radical | C61×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C61 ◁ C22×C4×C61 ◁ C2×#131×C61 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2)(3,15)(4,21)(5,25)(6,14)(7,12)(8,13)(9,17)(10,23)(11,28)(16,45)(18,38)(19,43)(20,44)(22,54)(24,41)(26,32)(27,40)(29,39)(30,37)(31,55)(33,50)(34,49)(35,60)(36,57)(42,47)(46,52)(48,62)(51,61)(53,56)(58,63)(59,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3,8,20)(2,9,13,33)(4,16,24,48)(5,26,29,49)(6,19,7,18)(10,35,37,31)(11,38,40,43)(12,34,14,32)(15,39,44,25)(17,27,50,28)(21,52,41,59)(22,55,57,60)(23,51,30,47)(36,62,54,45)(42,53,61,63)(46,58,64,56);(1,4)(2,10)(3,16)(5,22)(6,30)(7,23)(8,24)(9,31)(11,36)(12,41)(13,37)(14,21)(15,42)(17,46)(18,51)(19,47)(20,48)(25,53)(26,55)(27,58)(28,56)(29,57)(32,59)(33,35)(34,52)(38,45)(39,63)(40,54)(43,62)(44,61)(49,60)(50,64)2074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6299);