Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | H(5,𝔽2)×C97 ⋊ C4 |
| Sylow | C97 ⋊ #1402 |
122 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 1552 |
A minimal faithful permutation representation — 2 generators on 113 points:
(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(17,18,21,26,35,48,65,86,80,61,78,96,88,63,83,101,108,99,79,58,41,28,19,23,30,39,52,69,90,76,59,44,57,74,94,84,103,110,112,106,97,75,54,37,24,32,43,56,73,93,72,55,42,31,40,53,70,91,105,111,113,109,102,95,71,50,33,45,60,77,89,68,51,38,29,22,27,36,49,66,87,104,107,100,82,98,92,67,46,62,81,85,64,47,34,25,20);(1,2)(3,10,11,6)(4,8,14,7)(5,9)(12,16,13,15)(18,19,20,22)(21,24,25,31)(23,28,29,27)(26,33,34,44)(30,37,38,40)(32,41,42,36)(35,46,47,61)(39,50,51,57)(43,54,55,53)(45,58,59,49)(48,63,64,82)(52,67,68,78)(56,71,72,74)(60,75,76,70)(62,79,80,66)(65,84,85,102)(69,88,89,98)(73,92,93,96)(77,95,90,94)(81,97,86,91)(83,99,100,87)(101,106,107,105)(103,108,109,104)(110,112,113,111)Polycyclic presentation:
2^7,97|a^2=d,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^22,d^b=deg,e^b=eg,f^b=fg,d^c=dfg,e^c=eg,e^d=eg,h^d=h^96425 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19629);