Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12295 |
290 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 10112 |
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)(80,83,84,90)(81,86,87,93)(82,88,89,95)(85,91,92,94)(96,98)(97,103)(99,113)(100,110)(101,111)(102,117)(104,124)(105,121)(106,122)(107,116)(108,115)(109,114)(112,126)(118,120)(119,127)(123,125);(80,82)(81,85)(83,88)(84,89)(86,91)(87,92)(90,95)(93,94)(97,102)(99,112)(100,114)(103,117)(104,105)(106,119)(107,120)(108,125)(109,110)(113,126)(115,123)(116,118)(121,124)(122,127);(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)(80,81,84,87)(82,85,89,92)(83,86,90,93)(88,94,95,91)(96,97,101,106)(98,107,111,121)(99,102,115,119)(100,116,114,105)(103,110,122,109)(104,123,118,112)(108,120,113,124)(117,126,127,125)Polycyclic presentation:
2^8,79|a^2=g,c^2=gh,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,i^a=i^78,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=eh,i^c=i^781162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 82874);