Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13503 |
272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 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)(80,83,84,102,86,104,106,105)(81,89,90,116,93,117,92,122)(82,95,96,126,99,101,98,131)(85,107,87,103,108,137,111,138)(88,112,94,123,114,115,118,142)(91,120,119,133,121,136,109,100)(97,129,128,135,130,139,110,127)(113,134,124,141,140,125,143,132);(80,82)(81,88)(83,101)(84,98)(85,97)(86,99)(87,110)(89,115)(90,118)(91,113)(92,94)(93,114)(95,104)(96,106)(100,132)(102,126)(103,135)(105,131)(107,139)(108,130)(109,124)(111,128)(112,117)(116,123)(119,143)(120,125)(121,140)(122,142)(127,138)(129,137)(133,141)(134,136);(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)(80,81,87,109,106,90,85,91,86,93,111,119,84,92,108,121)(82,94,110,140,96,88,97,124,99,118,128,113,98,114,130,143)(83,100,107,117,102,136,103,116,104,133,137,89,105,120,138,122)(95,125,129,142,131,132,127,112,101,134,139,123,126,141,135,115)Polycyclic presentation:
2^8,79|a^2=eg,c^2=d,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^78,c^b=cf,d^b=dh,e^c=eg,f^c=fh,g^c=gh,i^c=i^781288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89493);