Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #2171 |
144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,9,42,92,67,21,8)(3,11,50,19,44,24,5,18)(4,17,52,88,89,82,30,15)(6,20,66,94,96,79,59,16)(7,23,74,57,93,81,43,41)(10,32,22,53,55,83,69,49)(12,29,73,77,61,85,35,39)(13,45,26,76,33,60,72,28)(14,31,34,64,56,63,25,51)(27,80,40,70,71,38,84,46)(36,68,97,62,58,65,75,37)(47,78,86,90,95,91,87,48)(98,101,103,111,104,116,121,106,105,117,122,124,102,115,120,125)(99,107,109,126,108,119,127,129,110,100,112,114,118,113,128,123);(2,4)(3,7)(5,10)(6,13)(8,16)(9,20)(11,25)(12,22)(14,28)(15,30)(17,34)(18,37)(19,40)(21,43)(23,26)(24,48)(27,44)(29,39)(31,56)(32,41)(33,38)(35,59)(36,55)(42,68)(45,65)(46,49)(47,54)(50,78)(51,71)(52,53)(57,61)(58,85)(60,87)(62,82)(63,81)(64,89)(66,77)(67,91)(69,70)(72,75)(73,84)(74,80)(76,86)(79,83)(88,95)(90,92)(93,94)(96,97)(98,100)(99,106)(101,112)(102,119)(103,114)(104,113)(105,107)(108,124)(109,117)(110,125)(111,118)(115,127)(116,128)(120,129)(121,123)(122,126);(2,3,5,8,4,7,10,16)(6,12,14,15,13,22,28,30)(9,19,21,41,20,40,43,32)(11,24,26,49,25,48,23,46)(17,33,35,18,34,38,59,37)(27,51,52,63,44,71,53,81)(29,45,54,72,39,65,47,75)(31,55,57,82,56,36,61,62)(42,67,69,80,68,91,70,74)(50,77,79,87,78,66,83,60)(58,84,86,64,85,73,76,89)(88,92,96,93,95,90,97,94)(98,99,103,109,104,108,121,127,105,110,122,112,102,118,120,128)(100,111,114,116,113,106,123,117,107,124,126,115,119,125,129,101)Polycyclic presentation:
2^7,97|a^2=e,c^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,c^a=cg,d^a=dg,h^a=h^64,d^b=dg,h^b=h^96,h^c=h^64,h^e=h^22,h^f=h^96128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21731);