Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #12884 |
302 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 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)(81,85)(84,87)(88,91,92,109,95,110,94,93)(89,97,98,107,100,99,111,115)(90,102,103,104,106,108,105,118)(96,112,101,116,113,114,117,119);(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)(80,82,83,86)(81,84,85,87)(88,90,92,103,95,106,94,105)(89,96,98,101,100,113,111,117)(91,108,109,118,110,102,93,104)(97,114,107,119,99,112,115,116);(80,81)(82,84)(83,85)(86,87)(88,89,93,99,94,98,110,115,95,100,109,97,92,111,91,107)(90,101,104,116,105,113,102,114,106,117,118,119,103,96,108,112)Polycyclic presentation:
2^8,79|a^2=f,b^2=e,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ch,i^b=i^78,i^c=i^782248 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85931);