Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C97 ⋊ D4×C16 |
174 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,12,58,95,90,29,11)(3,19,71,50,96,77,56,44)(4,13,61,28,59,32,5,26)(6,10,48,85,75,82,37,18)(7,30,57,94,92,87,45,9)(8,20,14,63,53,83,81,41)(15,39,67,79,51,74,23,55)(16,40,24,47,68,54,34,60)(17,38,33,70,22,49,65,36)(21,73,42,35,91,27,86,62)(25,84,93,52,69,66,72,46)(31,43,78,88,76,97,89,80)(102,104,105,111,106,112,113,117,107,103,108,109,114,110,115,116);(99,101);(2,3,5,9,6,4,8,11)(7,14,16,37,17,15,36,18)(10,21,23,46,24,22,45,26)(12,27,29,20,30,28,56,44)(13,31,33,42,34,32,19,41)(25,50,52,68,53,51,82,40)(35,62,48,77,59,60,63,54)(38,43,65,39,66,64,72,55)(47,69,67,88,75,74,73,70)(49,78,79,83,80,61,57,87)(58,89,81,71,84,90,86,91)(76,92,94,97,95,93,96,85)(98,99)(100,101)(102,103,105,109,106,110,113,116,107,104,108,111,114,112,115,117)Polycyclic presentation:
2^7,97|a^2=e,c^2=e,e^2=f,f^2=g,b^a=bd,h^a=h^33,h^c=h^64,h^e=h^22,h^f=h^96200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21718);