Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#33442 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C61 | C22×D4 |
| commutator | C4 | C23×C8×C61 |
| Frattini, cosocle | C2×C8 | C24×C61 |
| socle | C2×C61 | C2×C8×D4 |
| prime radical | C61×#575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C61 ◁ C2×M32×C61 ◁ G | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2,6,12,7,11,26,36,8,13,27,37,5,23,25,49)(3,14,17,39,16,9,28,30,18,31,43,55,24,32,50,56)(4,10,21,35,22,34,48,58,19,33,44,57,20,45,47,61)(15,38,42,59,41,29,51,53,40,54,60,64,46,52,62,63)(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)(2,9)(4,15)(5,24)(6,17)(7,16)(8,18)(10,29)(11,31)(12,30)(13,32)(14,23)(19,40)(20,46)(21,42)(22,41)(25,50)(26,28)(27,43)(33,52)(34,54)(35,53)(36,55)(37,56)(38,45)(39,49)(44,60)(47,62)(48,51)(57,63)(58,64)(59,61);(1,4,6,21,7,22,26,48,8,19,27,44,5,20,25,47)(2,10,12,35,11,34,36,58,13,33,37,57,23,45,49,61)(3,15,17,42,16,41,28,51,18,40,43,60,24,46,50,62)(9,29,30,53,31,54,55,64,32,52,56,63,14,38,39,59);(4,19)(10,33)(15,40)(20,22)(21,44)(29,52)(34,45)(35,57)(38,54)(41,46)(42,60)(47,48)(51,62)(53,63)(58,61)(59,64)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 26430);