Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | M32×C97 ⋊ C4 |
| Sylow | C97 ⋊ #1379 |
136 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 129 points:
(1,2,8,4)(3,6,18,14)(5,7,16,10)(9,12,32,24)(11,17,28,20)(13,15,30,26)(19,22,50,38)(21,31,42,34)(23,25,48,44)(27,29,46,40)(33,36,72,56)(35,49,60,52)(37,39,70,62)(41,47,66,58)(43,45,68,64)(51,54,89,78)(53,71,82,74)(55,57,88,83)(59,69,86,80)(61,63,75,87)(65,67,81,85)(73,76,84,91)(77,79,95,93)(90,92,94,97)(98,100)(99,105)(101,114)(102,116)(103,111)(104,113)(106,122)(107,124)(108,110)(109,121)(112,118)(115,128)(117,119)(120,126)(123,127)(125,129);(2,3,5,4)(6,9,11,10)(7,13,14,8)(12,19,21,20)(15,23,24,18)(16,17,27,26)(22,33,35,34)(25,37,38,32)(28,31,41,40)(29,43,44,30)(36,51,53,52)(39,55,56,50)(42,49,59,58)(45,61,62,48)(46,47,65,64)(54,73,75,74)(57,77,78,72)(60,71,81,80)(63,76,83,70)(66,69,86,85)(67,82,87,68)(79,90,91,89)(84,92,93,88)(94,96,97,95)(98,99,102,107,103,108,117,125,104,109,118,126,101,106,115,123)(100,110,112,127,111,121,128,124,113,122,116,129,114,105,119,120)Polycyclic presentation:
2^7,97|a^2=d,b^2=e,c^2=g,d^2=f,f^2=g,b^a=bc,h^a=h^75,c^b=cg,d^b=dg,h^b=h^22,h^d=h^96,h^e=h^96536 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19566);