Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C97 ⋊ #1749 |
412 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 6208 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2,5,10,19,32,49,70,63,45,62,80,73,48,67,85,92,84,64,43,26,13,4,7,14,23,36,53,74,59,42,28,41,58,78,69,87,94,97,91,81,60,39,22,9,16,27,40,57,76,55,38,25,15,24,37,54,75,89,95,96,93,86,79,56,35,18,29,44,61,72,51,34,21,12,6,11,20,33,50,71,88,90,83,66,82,77,52,31,46,65,68,47,30,17,8,3)(98,100,101,99)(107,110)(111,113);(2,4,3,6)(5,9,8,15)(7,13,12,11)(10,18,17,28)(14,22,21,24)(16,26,25,20)(19,31,30,45)(23,35,34,41)(27,39,38,37)(29,43,42,33)(32,48,47,66)(36,52,51,62)(40,56,55,58)(44,60,59,54)(46,64,63,50)(49,69,68,86)(53,73,72,82)(57,77,76,80)(61,79,74,78)(65,81,70,75)(67,84,83,71)(85,91,90,89)(87,92,93,88)(94,97,96,95)(103,105)(106,107,108,111,109,110,112,113);(2,3)(4,6)(5,8)(7,12)(9,15)(10,17)(11,13)(14,21)(16,25)(18,28)(19,30)(20,26)(22,24)(23,34)(27,38)(29,42)(31,45)(32,47)(33,43)(35,41)(36,51)(37,39)(40,55)(44,59)(46,63)(48,66)(49,68)(50,64)(52,62)(53,72)(54,60)(56,58)(57,76)(61,74)(65,70)(67,83)(69,86)(71,84)(73,82)(75,81)(77,80)(78,79)(85,90)(87,93)(88,92)(89,91)(94,96)(95,97)(98,99,101,100)(102,103)(104,105)Polycyclic presentation:
2^7,97|a^2=e,b^2=f,c^2=e,f^2=g,b^a=bd,h^a=h^96,c^b=cg,h^b=h^22,h^f=h^96824 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20237);