Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9008 |
398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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)(97,104)(98,108)(100,114)(101,113)(102,112)(103,116)(105,120)(106,111)(107,119)(109,125)(110,123)(115,124)(117,126)(118,127)(121,122);(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)(80,82,85,81)(83,90,92,87)(84,88,86,89)(91,94,93,95)(96,98,101,109,102,110,100,115)(97,103,106,117,107,118,105,121)(99,108,113,125,112,123,114,124)(104,116,111,126,119,127,120,122);(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)(96,97,101,106,102,107,100,105)(98,103,109,117,110,118,115,121)(99,111,114,104,112,120,113,119)(108,122,124,127,123,126,125,116)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,e^a=eg,i^a=i^78,d^b=dg,e^b=eh,f^b=fg,i^b=i^78,d^c=dh,e^c=egh,f^c=fh1288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 64533);