Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C79×#1287 ⋊ C2 |
| Sylow | C79 ⋊ #9334 |
398 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 398 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) = 127 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 127 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)(80,83,85,91)(81,86,87,90)(82,89,88,95)(84,93,94,92)(96,99)(97,104)(98,108)(100,112)(101,116)(102,113)(103,117)(105,120)(106,121)(107,122)(109,124)(110,126)(111,115)(114,125)(118,127)(119,123);(80,82,84,87,85,88,94,81)(83,89,93,90,91,95,92,86)(96,98,100,114,102,110,101,109)(97,103,105,118,106,119,115,107)(99,108,112,125,113,126,116,124)(104,117,120,127,121,123,111,122);(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)(80,81,84,88,85,87,94,82)(83,90,92,95,91,86,93,89)(96,97)(98,107,110,118)(99,111)(100,105)(101,115)(102,106)(103,109,119,114)(104,112)(108,123,126,117)(113,120)(116,121)(122,124,127,125)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^78,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=egh,f^c=fg,i^c=i^781282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 66541);