Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12171 |
332 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 332 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:
(1,4,7,20)(2,9)(3,14,17,43)(5,24,26,19)(6,21,25,54)(8,30)(10,33)(11,34)(12,18)(13,39)(15,46,47,42)(16,44,29,56)(22,57)(23,58,59,53)(27,60,45,55)(28,51)(31,49)(32,61)(35,62)(36,50)(37,63)(38,52)(40,41)(48,64)(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);(1,3,7,17)(2,8,12,31)(4,14,20,43)(5,23,26,59)(6,29,25,16)(9,30,18,49)(10,35,36,48)(11,37,28,32)(13,38,40,22)(15,45,47,27)(19,53,24,58)(21,56,54,44)(33,62,50,64)(34,63,51,61)(39,52,41,57)(42,60,46,55)(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)(3,13)(4,18)(5,22)(6,28)(7,12)(8,15)(9,20)(10,27)(11,25)(14,41)(16,48)(17,40)(19,52)(21,34)(23,37)(24,57)(26,38)(29,35)(30,46)(31,47)(32,59)(33,60)(36,45)(39,43)(42,49)(44,62)(50,55)(51,54)(53,61)(56,64)(58,63)Polycyclic presentation:
2^8,79|b^2=h,c^2=g,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=df,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, 82080);