Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12323 |
374 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(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)(81,86)(82,83)(85,87)(88,91,94,93)(89,96,98,108)(90,100,102,109)(92,103,107,106)(95,111,113,112)(97,104,115,116)(99,117,118,114)(101,105,110,119);(81,85)(82,83)(86,87)(88,90,94,102)(89,95,98,113)(91,100,93,109)(92,105,107,119)(96,111,108,112)(97,114,115,117)(99,116,118,104)(101,103,110,106);(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)(82,85)(83,86)(84,87)(88,89,94,98)(90,99,102,118)(91,96,93,108)(92,104,107,116)(95,110,113,101)(97,103,115,106)(100,117,109,114)(105,111,119,112)Polycyclic presentation:
2^8,79|a^2=h,b^2=h,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,i^a=i^78,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=eg1405 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 83041);