Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7832 |
322 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 127 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)(80,83)(81,86)(82,88)(84,90)(85,91)(87,93)(89,95)(92,94)(96,99,101,100)(97,103,104,108)(98,106,107,109)(102,110,111,105)(113,118)(115,123)(117,124)(119,120)(121,127)(125,126);(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)(81,85)(83,88)(84,89)(86,91)(87,92)(90,95)(93,94)(96,98,101,107)(97,102,104,111)(99,106,100,109)(103,110,108,105)(112,114,116,122)(113,117,120,126)(115,121,123,127)(118,124,119,125);(80,81,84,87)(82,85,89,92)(83,86,90,93)(88,94,95,91)(96,97,101,104)(98,105,107,110)(99,103,100,108)(102,109,111,106)(112,113,116,120)(114,117,122,126)(115,118,123,119)(121,124,127,125)Polycyclic presentation:
2^8,79|a^2=fgh,b^2=fh,c^2=f,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=df,e^b=eg,d^c=dg,e^c=eh,i^c=i^781285 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 58881);