Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C61 · Schur multiplier (H2) = C2 |
| direct product | C61×#7513 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C61 | C22⋊C4 |
| commutator | C22 | C2×C32×C61 |
| Frattini, cosocle | C2×C32 | C22×C61 |
| socle | C2×C61 | C22⋊C32 |
| prime radical | C23×C61 | C32 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C61 ◁ C2×C32×C61 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 2 generators on 125 points:
(1,2,3,9,4,10,14,27,5,11,15,28,17,30,36,49,6,12,16,29,18,31,37,50,20,32,38,51,39,52,56,62,7,8,13,23,19,24,33,43,21,25,34,44,40,46,53,58,22,26,35,45,41,47,54,59,42,48,55,60,57,61,63,64)(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);(2,8)(3,13)(10,24)(11,25)(12,26)(14,33)(15,34)(16,35)(30,46)(31,47)(32,48)(36,53)(37,54)(38,55)(52,61)(56,63)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 501);