Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C97 ⋊ #2960 |
364 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2)(3,8)(4,6)(5,7)(9,18)(10,12)(11,17)(13,16)(14,15)(19,32)(20,22)(21,31)(23,30)(24,25)(26,28)(27,29)(33,50)(34,36)(35,49)(37,48)(38,39)(40,42)(41,47)(43,46)(44,45)(51,72)(52,54)(53,71)(55,70)(56,57)(58,60)(59,69)(61,68)(62,63)(64,66)(65,67)(73,89)(74,76)(75,87)(77,88)(78,79)(80,82)(81,86)(83,84)(90,95)(91,92)(93,94)(96,97)(107,112)(111,117)(113,119)(118,121);(2,5,4,3)(6,11,10,9)(7,8,14,13)(12,21,20,19)(15,18,24,23)(16,27,26,17)(22,35,34,33)(25,32,38,37)(28,41,40,31)(29,30,44,43)(36,53,52,51)(39,50,56,55)(42,59,58,49)(45,48,62,61)(46,65,64,47)(54,75,74,73)(57,72,78,77)(60,81,80,71)(63,70,83,76)(66,86,85,69)(67,68,87,82)(79,89,91,90)(84,88,93,92)(94,95,97,96)(106,108,109,114,110,115,116,120)(107,111,113,118,112,117,119,121);(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)(98,100,101,105)(99,102,103,104);(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)(98,99,101,103)(100,104,105,102)(106,107,109,113,110,112,116,119)(108,111,114,118,115,117,120,121)Polycyclic presentation:
2^7,97|a^2=f,b^2=e,c^2=ef,f^2=g,b^a=be,d^a=dg,h^a=h^75,h^b=h^96,h^c=h^22,h^d=h^96,h^f=h^96392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21300);