Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q32×C97 ⋊ C4 |
| Sylow | C97 ⋊ #2180 |
142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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:
(1,2,8,5)(3,6,18,14)(4,7,16,11)(9,12,32,24)(10,17,28,21)(13,15,30,27)(19,22,50,38)(20,31,42,35)(23,25,48,44)(26,29,46,41)(33,36,72,56)(34,49,60,53)(37,39,70,62)(40,47,66,59)(43,45,68,65)(51,54,89,78)(52,71,82,75)(55,57,88,83)(58,69,85,81)(61,63,74,87)(64,67,80,86)(73,76,84,91)(77,79,95,93)(90,92,94,97)(98,101,103,102,104,115,120,117)(99,106,108,114,109,124,118,107)(100,111,119,116,112,128,113,129)(105,121,126,127,122,110,123,125);(2,4)(3,5)(6,10)(7,14)(8,13)(9,11)(12,20)(15,24)(16,26)(17,27)(18,23)(19,21)(22,34)(25,38)(28,40)(29,44)(30,43)(31,41)(32,37)(33,35)(36,52)(39,56)(42,58)(45,62)(46,64)(47,65)(48,61)(49,59)(50,55)(51,53)(54,74)(57,78)(60,80)(63,83)(66,85)(67,87)(68,82)(69,86)(70,76)(71,81)(72,77)(73,75)(79,91)(84,93)(88,92)(89,90)(94,97)(95,96)(98,100)(99,105)(101,111)(102,116)(103,119)(104,112)(106,121)(107,125)(108,126)(109,122)(110,124)(113,120)(114,127)(115,128)(117,129)(118,123);(2,3,4,5)(6,9,10,11)(7,13,14,8)(12,19,20,21)(15,23,24,18)(16,17,26,27)(22,33,34,35)(25,37,38,32)(28,31,40,41)(29,43,44,30)(36,51,52,53)(39,55,56,50)(42,49,58,59)(45,61,62,48)(46,47,64,65)(54,73,74,75)(57,77,78,72)(60,71,80,81)(63,76,83,70)(66,69,85,86)(67,82,87,68)(79,90,91,89)(84,92,93,88)(94,96,97,95)(98,99)(100,110,112,121)(101,114)(102,106)(103,118)(104,109)(105,111,122,128)(107,115)(108,120)(113,125,119,127)(116,126,129,123)(117,124)Polycyclic presentation:
2^7,97|a^2=e,c^2=ef,d^2=f,f^2=g,b^a=bd,c^a=cf,d^a=df,f^a=fg,h^a=h^75,d^b=dfg,e^b=eg,f^b=fg,h^b=h^96,d^c=dg,h^c=h^22,h^e=h^96287 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20917);