Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9104 |
474 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 474 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) = 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)(81,82)(86,87)(88,91,93,104)(89,95,97,102)(90,99,101,115)(92,106,107,103)(94,108,109,118)(96,111,105,110)(98,112,113,119)(100,116,117,114);(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)(80,82,84,87,83,81,85,86)(88,90,92,100,93,101,107,117)(89,94,96,98,97,109,105,113)(91,99,106,116,104,115,103,114)(95,108,111,112,102,118,110,119);(80,81,84,86,83,82,85,87)(88,89)(90,98)(91,102,104,95)(92,105)(93,97)(94,100)(96,107)(99,112,115,119)(101,113)(103,110,106,111)(108,114,118,116)(109,117)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,c^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,e^b=eg,i^b=i^78,d^c=dg,f^c=fg1762 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 65086);