Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C97 ⋊ #2164 |
138 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 3 generators on 137 points:
(1,2,6,15,31,58,82,93,79,61,68,40,66,86,46,74,71,24,50,72,54,26,5,11,27,43,70,87,59,84,65,12,20,7,17,36,63,88,96,67,92,89,33,52,14,29,37,16,32,41,8,18,39,28,53,38,64,90,91,97,95,81,55,34,3,9,22,45,73,35,60,69,21,44,25,10,23,47,75,78,48,51,80,49,76,77,19,42,57,83,94,62,85,56,30,13,4)(98,100,101,104)(99,103,102,105)(108,116)(112,122)(117,128)(118,129)(123,132)(124,133)(130,136)(134,137);(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)(99,102)(103,105)(106,108)(107,112)(109,117)(110,118)(111,116)(113,123)(114,124)(115,122)(119,130)(120,128)(121,129)(125,134)(126,132)(127,133)(131,136)(135,137);(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)(100,103)(101,102)(104,105)(106,107,109,113,110,114,119,125,111,115,120,126,121,127,131,135)(108,112,117,123,118,124,130,134,116,122,128,132,129,133,136,137)Polycyclic presentation:
2^7,97|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,h^a=h^64,c^b=cg,h^b=h^96,h^e=h^22,h^f=h^96176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21716);