Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #100×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)(19,27)(20,21)(22,23)(24,40)(25,26)(28,36)(29,30)(31,32)(33,57)(34,35)(37,53)(38,39)(41,49)(42,43)(44,45)(46,78)(47,48)(50,74)(51,52)(54,70)(55,56)(58,66)(59,60)(61,62)(63,98)(64,65)(67,96)(68,69)(71,94)(72,73)(75,91)(76,77)(79,87)(80,81)(82,83)(84,95)(85,86)(88,92)(89,90)(97,105)(99,104)(100,101)(102,103)(106,111)(107,108)(109,110)(112,113);(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,h^b=h^96,d^c=dfg,e^c=eg,e^d=eg,h^d=h^96281 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19630);