Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C79×#3202 ⋊ C2 |
| Sylow | C79 ⋊ #13352 |
682 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 682 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(1,4,7,21)(2,9,12,33)(3,14,17,44)(5,22,24,19)(6,26,27,20)(8,29,31,60)(10,34,36,32)(11,37,25,18)(13,38,40,62)(15,42,28,53)(16,47,48,43)(23,51,56,54)(30,61,57,49)(35,50,55,52)(39,63,46,41)(45,59,58,64)(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)(2,8)(4,14)(5,15)(6,16)(7,17)(9,29)(10,13)(11,30)(12,31)(18,49)(19,42)(20,43)(21,44)(22,53)(23,45)(24,28)(25,57)(26,47)(27,48)(32,38)(33,60)(34,62)(35,39)(36,40)(37,61)(41,50)(46,55)(51,64)(52,63)(54,59)(56,58)(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)(3,13,17,40)(4,18,21,37)(5,10,24,36)(6,25,27,11)(8,28,31,15)(9,26,33,20)(14,41,44,63)(16,46,48,39)(19,50,22,52)(23,55,56,35)(29,59,60,64)(30,45,57,58)(32,54,34,51)(38,47,62,43)(42,49,53,61)Polycyclic presentation:
2^8,79|a^2=h,c^2=h,e^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,g^b=gh,i^b=i^78,d^c=dg,e^c=eh,f^d=fh1042 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 88563);