Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #10614 |
276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 111 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)(81,87)(82,91)(84,85)(88,89)(92,93)(94,95)(96,99,101,98)(97,103,105,108)(100,107,106,109)(102,110,111,104);(80,82,83,92)(81,86,88,90)(84,91,85,93)(87,95,89,94)(96,98,106,109,101,99,100,107)(97,102,110,108,105,111,104,103);(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)(80,81,85,87,83,88,84,89)(82,90,93,94,92,86,91,95)(96,97,100,110,101,105,106,104)(98,103,107,102,99,108,109,111)Polycyclic presentation:
2^8,79|a^2=de,b^2=dg,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,i^a=i^78,c^b=cf,f^b=fh,d^c=dg,e^c=eh,f^c=fgh,i^c=i^781516 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 74265);