Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9374 |
368 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 368 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) = 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)(80,83,84,81)(82,90,93,87)(85,88,86,89)(91,94,92,95)(96,99,102,115)(97,104,106,112)(98,108,110,124)(100,114,101,113)(103,117,118,126)(105,120,116,119)(107,121,122,127)(109,125,111,123);(80,82,86,91,84,93,85,92)(81,87,88,95,83,90,89,94)(96,98,102,110)(97,103)(99,108,115,124)(100,109,101,111)(104,117)(105,107)(106,118)(112,126)(113,123,114,125)(116,122)(119,127)(120,121);(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)(80,81,86,89,84,83,85,88)(82,90,92,94,93,87,91,95)(96,97,101,105,102,106,100,116)(98,107,111,118,110,122,109,103)(99,112,114,120,115,104,113,119)(108,121,125,117,124,127,123,126)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,i^a=i^78,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fg,i^c=i^781285 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 66821);