Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25 |
| direct product | C22×C61×Q64 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C61 | D16 |
| commutator, Frattini, cosocle | C16 | C24×C61 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C22×C4×C61 ◁ C22×C8×C61 ◁ C22×C16×C61 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(3,4);(5,6,11,16)(7,17,21,48)(8,22,27,46)(9,28,31,62)(10,32,30,15)(12,34,37,18)(13,38,41,25)(14,42,44,24)(19,50,51,67)(20,52,33,47)(23,43,57,68)(26,60,59,54)(29,39,63,66)(35,61,64,56)(36,49,45,55)(40,58,53,65);(5,7,11,21)(6,12,16,37)(8,23,27,57)(9,29,31,63)(10,33,30,20)(13,39,41,66)(14,43,44,68)(15,45,32,36)(17,46,48,22)(18,38,34,25)(19,42,51,24)(26,61,59,56)(28,35,62,64)(40,67,53,50)(47,54,52,60)(49,58,55,65)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 26965);