Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12305 |
262 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,4)(2,9,12,18)(3,14)(5,19)(6,21)(7,20)(8,29,31,47)(10,36,37,33)(11,34,26,48)(13,39,40,57)(15,42)(16,44)(17,43)(22,49,54,61)(23,50)(24,52)(25,51)(27,53)(28,32)(30,58,38,62)(35,63,41,60)(45,56)(46,64)(55,59)(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)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)(130,131)(132,133)(134,135)(136,137)(138,139)(140,141)(142,143);(1,3,7,16)(2,8,12,31)(4,14,20,44)(5,23,24,45)(6,27,28,17)(9,29,18,47)(10,35,37,41)(11,38,26,30)(13,22,40,54)(15,25,46,55)(19,50,52,56)(21,53,32,43)(33,60,36,63)(34,62,48,58)(39,49,57,61)(42,51,64,59)(65,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,142,143,141,139,137,135,133,131,129,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67);(1,2,6,26,7,12,28,11)(3,13,17,35,16,40,27,41)(4,18,21,34,20,9,32,48)(5,22,25,10,24,54,55,37)(8,23,38,15,31,45,30,46)(14,39,43,63,44,57,53,60)(19,49,51,36,52,61,59,33)(29,56,62,64,47,50,58,42)Polycyclic presentation:
2^8,79|a^2=fg,b^2=h,c^2=g,d^2=f,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,i^c=i^78,e^d=eh805 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 82930);