Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #10123 |
442 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 79 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)(80,82)(81,85)(83,92)(84,89)(86,95)(87,94)(88,90)(91,93)(96,99,102,115)(97,104,106,121)(98,108,110,126)(100,114,101,113)(103,117,118,127)(105,123,116,120)(107,124,122,119)(109,112,111,125);(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)(82,85,89,94)(83,91,90,86)(88,95,92,93)(96,98,102,110)(97,103,106,118)(99,112,115,125)(100,111,101,109)(104,119,121,124)(105,122,116,107)(108,113,126,114)(117,123,127,120);(82,88)(83,90)(85,93)(86,91)(89,92)(94,95)(96,97)(98,107)(99,104)(100,116)(101,105)(102,106)(103,109)(108,119)(110,122)(111,118)(112,127)(113,123)(114,120)(115,121)(117,125)(124,126)Polycyclic presentation:
2^8,79|b^2=h,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cf,d^b=dg,e^b=eh,f^b=fg,i^b=i^78,d^c=dh,e^c=egh,f^c=fg,i^c=i^781279 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 71154);