Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | SD32×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:
(1,2,8,4)(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)(51,54,87,78)(53,71,82,74)(55,57,83,84)(59,69,86,80)(61,63,77,79)(65,67,73,76)(75,89,92,85)(81,88,94,90)(91,95,96,93)(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);(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);(2,3,5,4)(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)(63,83,84,70)(66,69,85,76)(67,87,79,68)(82,89,91,90)(86,88,93,92)(94,95,97,96)(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^22,d^b=dfg,e^b=eg,f^b=fg,d^c=dg,h^c=h^75,h^e=h^96287 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20916);