Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16×C61 · Schur multiplier (H2) = C25 |
| direct product | C2×C61×#2258 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C61 | C22 |
| commutator | C2 | C23×C16×C61 |
| Frattini, cosocle | C16 | C24×C61 |
| socle | C22×C61 | C22×C16 |
| prime radical | C2×#41×C61 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C61 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(3,4,6,11,7,12,19,29,8,13,20,30,22,32,41,51,9,14,21,31,23,33,42,52,24,34,43,53,44,54,59,64)(5,10,16,26,17,27,38,48,18,28,39,49,40,50,58,63,15,25,35,45,36,46,55,60,37,47,56,61,57,62,65,66);(3,5)(4,10)(6,16)(7,17)(8,18)(9,15)(11,26)(12,27)(13,28)(14,25)(19,38)(20,39)(21,35)(22,40)(23,36)(24,37)(29,48)(30,49)(31,45)(32,50)(33,46)(34,47)(41,58)(42,55)(43,56)(44,57)(51,63)(52,60)(53,61)(54,62)(59,65)(64,66);(5,15)(10,25)(16,35)(17,36)(18,37)(26,45)(27,46)(28,47)(38,55)(39,56)(40,57)(48,60)(49,61)(50,62)(58,65)(63,66)9760 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 26961);