Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#13392 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C2×C61 | #2033 |
| prime radical | C61×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C22×C4×C61 ◁ C2×#131×C61 ◁ G | |
386 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)(3,13)(4,19)(5,25)(6,28)(7,12)(8,15)(9,22)(10,27)(11,26)(14,44)(16,33)(17,34)(18,35)(20,54)(21,52)(23,53)(24,40)(29,39)(30,37)(31,58)(32,60)(36,57)(38,56)(41,59)(42,46)(43,49)(45,61)(47,63)(48,62)(50,51)(55,64)(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)(2,8)(4,20)(5,17)(6,29)(7,18)(9,32)(10,35)(11,39)(12,34)(13,25)(14,23)(15,27)(16,31)(19,51)(21,45)(22,46)(24,50)(26,47)(28,43)(30,48)(33,41)(36,64)(37,42)(38,62)(40,44)(49,59)(52,54)(53,61)(55,56)(57,60)(58,63);(1,4,5,21,7,23,27,24)(2,9,10,36,12,38,25,37)(3,14,15,45,18,20,17,50)(6,30,26,56,31,57,59,22)(8,32,13,42,34,62,35,64)(11,40,28,53,41,52,58,19)(16,48,47,46,29,60,49,55)(33,54,43,61,39,44,63,51)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6380);