Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#24989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C61 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×#41×C61 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C61 ◁ C22×C4×C61 ◁ G | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2,8,14)(3,15,20,33)(4,21,25,40)(5,26,29,38)(6,13,7,12)(9,19,34,18)(10,30,37,24)(11,28,39,27)(16,35,22,31)(17,45,48,32)(23,53,55,36)(41,49,50,44)(42,47,58,46)(43,62,52,61)(51,56,60,54)(57,63,59,64)(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);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,24,49,25,16,30,44)(5,17,28,47,29,48,27,46)(10,35,21,50,37,31,40,41)(11,32,38,58,39,45,26,42)(23,52,54,64,55,43,56,63)(36,59,51,62,53,57,60,61);(1,4)(2,10)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31)(11,36)(12,21)(13,40)(14,37)(15,41)(17,43)(18,44)(19,49)(20,22)(26,51)(27,54)(28,56)(29,55)(32,57)(33,50)(34,35)(38,60)(39,53)(42,61)(45,59)(46,63)(47,64)(48,52)(58,62);(1,5,7,28,8,29,6,27)(2,11,13,38,14,39,12,26)(3,17,19,47,20,48,18,46)(4,23,30,56,25,55,24,54)(9,32,33,58,34,45,15,42)(10,36,40,60,37,53,21,51)(16,43,49,64,22,52,44,63)(31,57,50,62,35,59,41,61)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 17977);