Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q64×C97 ⋊ C2 |
| Sylow | C97 ⋊ #2264 |
104 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 3 generators on 161 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)(99,106)(105,127)(107,130)(108,131)(109,125)(110,137)(115,136)(121,155)(128,157)(129,135)(132,158)(133,146)(134,159)(138,145)(142,161)(147,153);(98,100,104,114)(99,105,106,127)(101,116,120,150)(102,122,124,156)(103,126,123,113)(107,132,130,158)(108,134,131,159)(109,135,125,129)(110,136,137,115)(111,139,141,118)(112,143,144,117)(119,154,152,149)(121,128,155,157)(133,161,146,142)(138,147,145,153)(140,151,148,160);(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)(98,99)(100,110)(101,115)(102,121)(103,125)(104,106)(105,111)(107,118)(108,117)(109,123)(112,142)(113,145)(114,137)(116,134)(119,153)(120,136)(122,132)(124,155)(126,138)(127,141)(128,149)(129,148)(130,139)(131,143)(133,160)(135,140)(144,161)(146,151)(147,152)(150,159)(154,157)(156,158)Polycyclic presentation:
2^7,97|b^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,h^a=h^96,d^b=de,e^b=ef,f^b=fg,h^c=h^96938 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 6829);