Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9422 |
384 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 99 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)(80,83,85,92)(81,87,82,90)(84,93,86,91)(88,95,89,94)(96,99,102,112)(97,104,105,120)(98,107,108,123)(100,114,101,111)(103,116,119,126)(106,121,113,110)(109,125,115,124)(117,127,118,122);(80,82,85,81)(83,90,92,87)(84,88,86,89)(91,94,93,95)(96,98,100,109,102,108,101,115)(97,103,106,117,105,119,113,118)(99,107,114,125,112,123,111,124)(104,116,121,127,120,126,110,122);(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)(80,81,86,89,85,82,84,88)(83,87,93,95,92,90,91,94)(96,97,101,106,102,105,100,113)(98,103,109,118,108,119,115,117)(99,110,111,104,112,121,114,120)(107,122,125,126,123,127,124,116)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfh,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^78,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eg,f^c=fgh1516 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 67090);