Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| Sylow | C79 ⋊ #8794 |
270 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(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)(80,82,83,85,84,86,87,81)(88,91,92,103,93,104,105,118)(89,95,96,108,97,102,109,117)(90,99,100,113,101,114,115,119)(94,106,98,110,107,116,111,112);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,81,83,82,84,85,87,86)(88,90)(89,94,97,107)(91,99)(92,100)(93,101)(95,106,102,116)(96,98,109,111)(103,113)(104,114)(105,115)(108,110,117,112)(118,119);(80,81,84,85)(82,83,86,87)(88,89)(90,98,101,111)(91,102,104,95)(92,96)(93,97)(94,100,107,115)(99,112)(103,117,118,108)(105,109)(106,113)(110,114)(116,119)Polycyclic presentation:
2^8,79|a^2=f,b^2=e,c^2=d,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,f^b=fgh,f^c=fgh,i^c=i^781288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 63263);