Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×Q64×C79 ⋊ C2 |
| Sylow | C79 ⋊ #13672 |
274 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 207 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 207 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)(80,83,87,106)(81,89,93,127)(82,95,100,143)(84,102,112,159)(85,105,114,103)(86,104,117,162)(88,119,123,178)(90,124,131,185)(91,126,113,101)(92,125,134,187)(94,135,138,192)(96,140,118,175)(97,146,99,141)(98,142,149,198)(107,163,166,200)(108,154,169,203)(109,155,132,160)(110,158,172,156)(111,157,128,188)(115,174,173,152)(116,161,147,199)(120,179,122,150)(121,177,181,194)(129,183,191,197)(130,184,165,151)(133,186,180,206)(136,193,137,139)(144,176,145,196)(148,195,190,207)(153,164,201,171)(167,205,168,202)(170,182,189,204);(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)(80,82)(81,88)(83,95)(84,108)(85,97)(86,116)(87,100)(89,119)(90,129)(91,120)(92,133)(93,123)(94,107)(96,111)(98,109)(99,114)(101,150)(102,154)(103,141)(104,161)(105,146)(106,143)(110,167)(112,169)(113,122)(115,121)(117,147)(118,128)(124,183)(125,186)(126,179)(127,178)(130,190)(131,191)(132,149)(134,180)(135,163)(136,164)(137,171)(138,166)(139,153)(140,157)(142,155)(144,170)(145,189)(148,165)(151,195)(152,194)(156,202)(158,205)(159,203)(160,198)(162,199)(168,172)(173,181)(174,177)(175,188)(176,182)(184,207)(185,197)(187,206)(192,200)(193,201)(196,204);(80,81,87,93)(82,94,100,138)(83,101,106,126)(84,107,112,166)(85,113,114,91)(86,115,117,173)(88,118,123,96)(89,105,127,103)(90,128,131,111)(92,132,134,109)(95,139,143,193)(97,137,99,136)(98,148,149,190)(102,153,159,201)(104,125,162,187)(108,133,169,180)(110,171,172,164)(116,129,147,191)(119,176,178,196)(120,144,122,145)(121,167,181,168)(124,182,185,204)(130,170,165,189)(135,146,192,141)(140,150,175,179)(142,197,198,183)(151,188,184,157)(152,160,174,155)(154,194,203,177)(156,163,158,200)(161,195,199,207)(186,205,206,202)Polycyclic presentation:
2^8,79|a^2=h,c^2=h,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^b=i^78,e^c=eh,i^c=i^781519 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90272);