Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #2169 |
162 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2,9,42,92,67,20,7)(3,11,50,19,44,24,4,17)(5,18,52,90,87,82,30,15)(6,21,66,94,96,78,59,16)(8,23,74,56,93,81,43,40)(10,32,22,53,55,83,68,48)(12,29,73,77,63,86,38,36)(13,31,33,60,57,65,26,51)(14,45,25,76,37,62,72,28)(27,80,41,70,71,34,84,46)(35,39,69,95,64,58,61,75)(47,79,85,88,97,91,89,49)(102,104,106,113,107,108,111,116,105,112,114,117,115,103,109,110);(99,101)(103,108)(104,112)(110,116)(113,117);(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)(100,101)(102,103,106,110,107,104,111,113,105,108,114,116,115,112,109,117)Polycyclic presentation:
2^7,97|a^2=e,c^2=e,e^2=f,f^2=g,b^a=bd,c^a=cg,h^a=h^64,c^b=cg,h^c=h^64,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, 21724);