Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #10227 |
466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,82,84,81)(83,85,87,86)(88,91)(89,96)(90,100)(92,107)(93,106)(94,105)(95,109)(97,114)(98,103)(99,113)(101,117)(102,116)(104,108)(110,119)(111,118)(112,115);(80,81,84,82)(83,86,87,85)(88,90)(89,95)(91,104)(92,101)(93,108)(94,102)(96,112)(97,110)(98,115)(99,111)(100,107)(103,118)(105,117)(106,116)(109,114)(113,119);(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)(81,85)(82,86)(83,87)(88,89,94,99)(90,95,102,111)(91,103,105,114)(92,98,93,97)(96,107,113,106)(100,112,116,119)(101,115,108,110)(104,109,117,118)Polycyclic presentation:
2^8,79|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,i^a=i^78,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,d^c=dh,e^c=eh,f^c=fgh1282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 71860);