Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#20261787 |
| Sylow | C97 ⋊ C2×#446 |
400 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2,8,5)(3,6,18,14)(4,7,16,11)(9,12,32,24)(10,17,28,21)(13,15,30,27)(19,22,50,38)(20,31,42,35)(23,25,48,44)(26,29,46,41)(33,36,72,56)(34,49,60,53)(37,39,70,62)(40,47,66,59)(43,45,68,65)(51,54,89,78)(52,71,82,75)(55,57,88,83)(58,69,85,81)(61,63,74,87)(64,67,80,86)(73,76,84,91)(77,79,95,93)(90,92,94,97)(108,110,111,113,112,114,115,109);(2,4)(3,5)(6,10)(7,14)(8,13)(9,11)(12,20)(15,24)(16,26)(17,27)(18,23)(19,21)(22,34)(25,38)(28,40)(29,44)(30,43)(31,41)(32,37)(33,35)(36,52)(39,56)(42,58)(45,62)(46,64)(47,65)(48,61)(49,59)(50,55)(51,53)(54,74)(57,78)(60,80)(63,83)(66,85)(67,87)(68,82)(69,86)(70,76)(71,81)(72,77)(73,75)(79,91)(84,93)(88,92)(89,90)(94,97)(95,96)(100,102,103,107)(101,104,105,106);(2,3,4,5)(6,9,10,11)(7,13,14,8)(12,19,20,21)(15,23,24,18)(16,17,26,27)(22,33,34,35)(25,37,38,32)(28,31,40,41)(29,43,44,30)(36,51,52,53)(39,55,56,50)(42,49,58,59)(45,61,62,48)(46,47,64,65)(54,73,74,75)(57,77,78,72)(60,71,80,81)(63,76,83,70)(66,69,85,86)(67,82,87,68)(79,90,91,89)(84,92,93,88)(94,96,97,95)(100,101,103,105)(102,106,107,104)(108,109,111,110,112,113,115,114);(98,99)536 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21297);