Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #408×C97 ⋊ C2 |
| Sylow | C97 ⋊ #2017 |
350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 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)(100,110)(102,116)(103,117)(104,118)(105,120)(107,124)(108,125)(109,126)(111,115)(112,129)(113,128)(114,119)(121,123)(122,127);(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)(98,100)(99,105)(101,115)(102,112)(103,111)(104,113)(106,123)(107,121)(108,122)(109,119)(110,116)(114,120)(117,128)(118,129)(124,127)(125,126);(98,99)(100,109)(101,114)(102,119)(103,107)(104,108)(105,111)(106,117)(110,127)(112,122)(113,121)(115,126)(116,125)(118,124)(120,128)(123,129)Polycyclic presentation:
2^7,97|d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,f^b=fg,h^b=h^96,e^c=eg,f^c=fg,h^c=h^961370 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 5614);