Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#33974 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C61 | C24 |
| commutator | C2 | C23×C16×C61 |
| Frattini, cosocle | C16 | C24×C61 |
| socle | C2×C61 | C23×C16 |
| prime radical | C2×#41×C61 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C61 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2,4,10,5,11,17,30,6,12,18,31,20,32,39,52,7,9,19,27,21,28,40,49,22,29,41,50,42,51,57,62)(3,8,14,24,15,25,36,46,16,26,37,47,38,48,56,61,13,23,33,43,34,44,53,58,35,45,54,59,55,60,63,64)(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)(4,14)(5,15)(6,16)(7,13)(9,23)(10,24)(11,25)(12,26)(17,36)(18,37)(19,33)(20,38)(21,34)(22,35)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(39,56)(40,53)(41,54)(42,55)(49,58)(50,59)(51,60)(52,61)(57,63)(62,64);(3,13)(8,23)(14,33)(15,34)(16,35)(24,43)(25,44)(26,45)(36,53)(37,54)(38,55)(46,58)(47,59)(48,60)(56,63)(61,64);(2,9)(8,23)(10,27)(11,28)(12,29)(24,43)(25,44)(26,45)(30,49)(31,50)(32,51)(46,58)(47,59)(48,60)(52,62)(61,64)8296 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 26962);