Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #10669 |
296 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 89 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)(80,82,85,93)(81,87,89,95)(83,92,84,91)(86,94,88,90)(96,99)(97,103)(98,106)(100,112)(101,110)(102,114)(104,119)(105,118)(107,124)(108,122)(109,120)(111,113)(115,126)(116,117)(121,127)(123,125);(81,86)(82,91)(83,84)(87,95)(88,89)(92,93)(96,98,100,107,101,108,113,125)(97,102,104,115,105,116,120,127)(99,106,112,124,110,122,111,123)(103,117,119,121,118,114,109,126);(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)(80,81,84,86,85,89,83,88)(82,90,92,95,93,94,91,87)(96,97)(98,102)(99,109,110,119)(100,104)(101,105)(103,111,118,112)(106,121,122,126)(107,115)(108,116)(113,120)(114,123,117,124)(125,127)Polycyclic presentation:
2^8,79|a^2=dg,b^2=de,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,f^a=fgh,i^a=i^78,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eh,f^c=fg,i^c=i^781756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 74596);