Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C97 ⋊ #1369 |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 2 generators on 129 points:
(1,2,6,12,29,56,53,79,89,54,81,38,62,88,49,74,69,23,51,70,45,25,10,22,48,34,61,75,18,41,67,27,35,15,32,59,73,91,97,85,95,65,19,43,30,46,20,8,17,33,3,9,21,13,31,42,68,90,92,94,96,82,77,39,7,16,36,52,66,40,63,86,37,58,24,4,11,26,44,71,50,57,83,47,72,87,60,84,80,76,93,78,64,55,28,14,5)(98,100,101,112,103,114,118,113,105,116,119,127,104,115,102,120)(99,106,107,111,109,122,117,126,110,123,108,124,121,129,125,128);(2,3,4,7,5,8,10,15)(6,12,13,27,14,28,30,36)(9,18,19,39,20,40,42,32)(11,23,24,49,25,50,22,47)(16,33,34,60,35,17,37,38)(21,44,45,67,46,70,26,52)(29,53,54,77,55,78,80,59)(31,57,58,82,43,69,48,73)(41,64,65,88,66,56,68,72)(51,75,76,87,71,63,89,62)(61,85,81,74,86,92,84,83)(79,94,95,96,93,97,90,91)(98,99,102,108,104,109,119,125,105,110,118,107,103,121,101,117)(100,111,113,122,115,126,112,123,116,124,120,129,114,128,127,106)Polycyclic presentation:
2^7,97|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,h^a=h^50,d^b=dg,h^d=h^75,h^f=h^96224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21573);