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:
(1,2,6,15,31,58,82,93,78,63,69,41,66,85,46,74,71,24,50,72,54,25,4,11,27,43,70,89,59,84,61,12,21,8,18,39,65,90,96,67,92,87,37,52,13,29,35,16,32,40,7,17,36,28,53,34,60,88,91,95,97,81,55,33,3,9,22,45,73,38,62,68,20,44,26,10,23,47,75,79,49,51,80,48,76,77,19,42,56,83,94,64,86,57,30,14,5)(98,100,101,104)(99,103,102,105)(108,116)(112,122)(117,128)(118,129)(123,132)(124,133)(130,136)(134,137);(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);(2,3,4,7,5,8,10,16)(6,12,13,15,14,22,28,30)(9,19,20,40,21,41,43,32)(11,24,25,48,26,49,23,46)(17,33,34,59,35,18,37,38)(27,51,52,65,44,71,53,81)(29,45,54,72,36,61,47,75)(31,55,56,82,57,39,63,64)(42,67,68,80,69,91,70,74)(50,77,78,89,79,66,83,62)(58,84,85,60,86,73,76,87)(88,95,94,90,92,96,93,97)(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^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, 21715);