Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #12892 |
274 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 10112 |
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)(81,85)(84,87)(88,91,92,107,111,104,93,106,95,109,113,105,94,108,112,90)(89,97,98,117,100,119,99,96,102,103,114,116,101,115,110,118);(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)(80,82,83,86)(81,84,85,87)(88,90,93,104,94,105,92,91,95,106,112,108,111,107,113,109)(89,96,99,115,101,117,98,103,102,118,110,119,100,116,114,97);(80,81)(82,84)(83,85)(86,87)(88,89,94,100,95,102,111,101)(90,103,105,115,106,97,107,119)(91,96,108,116,109,118,104,117)(92,110,112,98,113,99,93,114)Polycyclic presentation:
2^8,79|a^2=f,b^2=e,c^2=d,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,c^b=ch,i^c=i^781768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85959);