Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #472×C97 ⋊ C2 |
| Sylow | C97 ⋊ #3290 |
454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(98,101)(99,106)(100,111)(102,117)(103,118)(104,122)(105,119)(107,126)(108,125)(109,128)(110,112)(113,115)(114,129)(116,120)(121,124)(123,127);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,101)(99,105)(100,110)(102,116)(103,118)(104,121)(106,119)(107,125)(108,126)(109,128)(111,112)(113,129)(114,115)(117,120)(122,124)(123,127);(98,100)(99,104)(101,115)(102,113)(103,114)(105,122)(106,127)(107,123)(108,124)(109,119)(110,118)(111,116)(112,120)(117,129)(121,126)(125,128);(98,99)(100,109)(101,106)(102,119)(103,108)(104,112)(105,117)(107,120)(110,122)(111,128)(113,124)(114,123)(115,121)(116,126)(118,125)(127,129)Polycyclic presentation:
2^7,97|e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,c^b=cg,d^b=de,e^b=eg,f^c=fg,h^c=h^96,e^d=eg,h^d=h^96797 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 16291);