Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12302 |
310 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 310 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) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,31)(11,37)(12,39)(13,21)(14,35)(15,42)(17,46)(18,51)(19,47)(20,48)(25,58)(26,54)(27,56)(28,57)(30,55)(32,61)(33,62)(34,38)(36,63)(40,53)(41,52)(43,45)(44,60)(49,64)(50,59)(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);(1,3,8,20)(2,9,14,33)(4,16,24,48)(5,26,28,49)(6,19,7,18)(10,31,35,62)(11,36,38,43)(12,32,13,41)(15,25,44,40)(17,27,50,30)(21,52,39,61)(22,54,57,64)(23,51,29,47)(34,45,37,63)(42,58,60,53)(46,56,59,55)(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);(1,2,7,12,8,14,6,13)(3,15,19,36,20,44,18,43)(4,21,23,10,24,39,29,35)(5,25,27,11,28,40,30,38)(9,26,41,17,33,49,32,50)(16,45,47,42,48,63,51,60)(22,37,56,53,57,34,55,58)(31,59,52,54,62,46,61,64)Polycyclic presentation:
2^8,79|a^2=fg,b^2=h,d^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,i^b=i^78,d^c=dg,e^c=eh,e^d=eh1039 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 82914);