Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C97 ⋊ #2117 |
220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 3 generators on 129 points:
(1,2,6,15,31,58,82,93,78,63,69,41,66,85,46,74,71,24,50,72,54,25,4,11,27,43,70,89,59,84,61,12,21,8,18,39,65,90,96,67,92,87,37,52,13,29,35,16,32,40,7,17,36,28,53,34,60,88,91,95,97,81,55,33,3,9,22,45,73,38,62,68,20,44,26,10,23,47,75,79,49,51,80,48,76,77,19,42,56,83,94,64,86,57,30,14,5)(99,106)(105,120)(107,109)(108,122)(110,123)(121,125)(124,126)(127,129);(98,100)(99,105)(101,114)(102,116)(103,111)(104,113)(106,120)(107,123)(108,125)(109,110)(112,118)(115,128)(117,119)(121,122)(124,127)(126,129);(2,3,4,7,5,8,10,16)(6,12,13,15,14,22,28,30)(9,19,20,40,21,41,43,32)(11,24,25,48,26,49,23,46)(17,33,34,59,35,18,37,38)(27,51,52,65,44,71,53,81)(29,45,54,72,36,61,47,75)(31,55,56,82,57,39,63,64)(42,67,68,80,69,91,70,74)(50,77,78,89,79,66,83,62)(58,84,85,60,86,73,76,87)(88,95,94,90,92,96,93,97)(98,99,102,108,103,109,117,126,104,106,118,122,101,107,115,124)(100,110,112,127,111,120,128,125,113,123,116,129,114,105,119,121)Polycyclic presentation:
2^7,97|a^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,c^a=cg,h^a=h^64,d^b=dg,e^b=eg,h^e=h^22,h^f=h^96164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21692);