Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊#15690909 |
| Sylow | C97 ⋊ C22⋊C32 |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 2 generators on 133 points:
(1,2,7,21,30,73,70,65,3,11,32,75,22,53,61,94,17,42,25,59,77,43,5,16,45,67,74,8,24,58,55,18,38,13,35,36,83,63,92,87,64,62,82,84,19,50,56,33,76,66,12,34,80,49,46,79,90,97,60,95,96,47,69,78,4,14,39,85,93,57,23,54,28,68,44,15,40,86,81,9,26,48,89,88,91,52,37,41,31,10,27,29,71,72,51,20,6)(99,101);(2,3,4,8,5,9,12,22,6,10,13,23,15,25,33,52)(7,17,18,46,19,47,21,45,20,48,39,84,49,83,51,68)(11,28,29,66,30,38,69,75,31,67,70,76,72,14,36,37)(16,41,42,87,43,54,88,89,44,32,26,60,40,74,61,94)(24,55,56,86,35,81,82,50,57,85,34,77,78,59,79,80)(27,62,63,73,64,58,92,53,65,90,96,71,97,93,95,91)(98,99)(100,101)(102,103,104,108,105,109,112,118,106,110,113,119,115,121,124,128,107,111,114,120,116,122,125,129,117,123,126,130,127,131,132,133)Polycyclic presentation:
2^7,97|a^2=d,d^2=e,e^2=f,f^2=g,b^a=bc,h^a=h^8,h^d=h^64,h^e=h^22,h^f=h^96128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21781);