Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | #107×C97 ⋊ C4 |
| Sylow | C97 ⋊ #1772 |
232 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 3104 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2,8,4)(3,6,18,14)(5,7,16,10)(9,12,32,24)(11,17,28,20)(13,15,30,26)(19,22,50,38)(21,31,42,34)(23,25,48,44)(27,29,46,40)(33,36,72,56)(35,49,60,52)(37,39,70,62)(41,47,66,58)(43,45,68,64)(51,54,89,78)(53,71,82,74)(55,57,88,83)(59,69,86,80)(61,63,75,87)(65,67,81,85)(73,76,84,91)(77,79,95,93)(90,92,94,97)(102,104,105,110,106,111,113,117)(103,108,112,116,109,115,107,114);(98,99,100,101)(102,103)(104,108)(105,112)(106,109)(107,113)(110,116)(111,115)(114,117);(2,3,5,4)(6,9,11,10)(7,13,14,8)(12,19,21,20)(15,23,24,18)(16,17,27,26)(22,33,35,34)(25,37,38,32)(28,31,41,40)(29,43,44,30)(36,51,53,52)(39,55,56,50)(42,49,59,58)(45,61,62,48)(46,47,65,64)(54,73,75,74)(57,77,78,72)(60,71,81,80)(63,76,83,70)(66,69,86,85)(67,82,87,68)(79,90,91,89)(84,92,93,88)(94,96,97,95)(103,107,109,112)(108,114,115,116)Polycyclic presentation:
2^7,97|a^2=e,b^2=f,c^2=de,d^2=g,b^a=bd,h^a=h^75,d^b=dg,e^b=eg,h^c=h^22,h^e=h^96392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20314);