Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C2×C22⋊C4×C97 ⋊ C4 |
| Sylow | C97 ⋊ C4×#99 |
412 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2)(3,5)(4,11)(6,7)(8,10)(9,24)(12,14)(13,20)(15,16)(17,19)(18,41)(21,23)(22,37)(25,27)(26,33)(28,29)(30,32)(31,62)(34,36)(35,58)(38,40)(39,54)(42,44)(43,50)(45,46)(47,49)(48,82)(51,53)(52,80)(55,57)(56,78)(59,61)(60,75)(63,65)(64,71)(66,67)(68,70)(69,79)(72,74)(73,77)(81,89)(83,85)(84,88)(86,87)(90,92)(91,95)(93,94)(96,97)(98,100,101,99);(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)(102,104)(103,106,107,108)(105,109);(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)(104,108)(105,107)(106,109)Polycyclic presentation:
2^7,97|a^2=e,b^2=f,c^2=e,b^a=bd,f^a=fg,h^a=h^96,d^b=dg,h^b=h^22,h^c=h^96,h^f=h^96524 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20257);