Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12400 |
286 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 10112 |
A minimal faithful permutation representation — 3 generators on 127 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)(81,87)(83,84)(86,94)(88,89)(90,95)(91,92)(99,112)(103,122)(106,125)(107,117)(110,124)(111,127)(114,120)(121,126);(80,82,85,93)(81,86,89,95)(83,92,84,91)(87,94,88,90)(96,98,101,108)(97,102,105,118)(99,111,112,127)(100,116,113,109)(103,121,122,126)(104,123,115,119)(106,124,125,110)(107,120,117,114);(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)(80,81,85,89)(82,90,93,94)(83,88,84,87)(86,92,95,91)(96,97,101,105)(98,106,108,125)(99,110,112,124)(100,115,113,104)(102,117,118,107)(103,120,122,114)(109,126,116,121)(111,123,127,119)Polycyclic presentation:
2^8,79|a^2=g,b^2=g,d^2=fh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,f^a=fh,i^a=i^78,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=egh1162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 83572);