Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#13016 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | #447 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C2×C61 | #1879 |
| prime radical | C2×C61×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C22×C4×C61 ◁ C61×#405 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2,7,13,8,14,30,46,9,15,6,31,29,45,5,12)(3,16,20,39,21,44,18,47,22,50,19,10,34,37,32,38)(4,23,25,48,26,59,24,11,27,41,28,42,35,40,33,43)(17,51,53,58,54,36,56,49,55,62,60,64,57,63,52,61)(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);(1,3,8,34,9,22,29,21)(2,10,14,47,15,39,45,38)(4,17,35,54,27,55,26,57)(5,19,7,18,30,20,6,32)(11,36,48,62,43,63,42,51)(12,44,13,16,46,37,31,50)(23,58,40,49,41,64,59,61)(24,53,25,56,33,60,28,52);(1,4)(2,11)(3,17)(5,28)(6,33)(7,24)(8,35)(9,27)(10,36)(12,23)(13,40)(14,48)(15,43)(16,49)(18,53)(19,52)(20,56)(21,57)(22,55)(25,30)(26,29)(31,59)(32,60)(34,54)(37,64)(38,51)(39,63)(41,46)(42,45)(44,58)(47,62)(50,61)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6004);