Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #13544 |
212 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 212 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) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 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)(80,83,84,100,85,101,87,109)(81,89,90,114,93,116,92,123)(82,95,96,126,97,127,108,128)(86,106,103,134,104,125,107,102)(88,110,94,124,112,138,122,137)(91,119,117,141,121,143,120,115)(98,130,105,99,129,132,135,133)(111,139,118,113,136,131,142,140);(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)(80,82,87,108,85,97,84,96)(81,88,90,94,93,112,92,122)(83,99,109,130,101,133,100,132)(86,105,107,98,104,135,103,129)(89,113,114,131,116,140,123,139)(91,118,117,136,121,142,120,111)(95,125,128,134,127,106,126,102)(110,119,124,141,138,143,137,115);(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)(82,94,97,122)(83,89,101,116)(84,90)(85,93)(86,91)(87,92)(88,96,112,108)(95,124)(98,118,129,142)(99,131)(100,114,109,123)(102,115,134,141)(103,117)(104,121)(105,136,135,111)(106,119,125,143)(107,120)(110,128)(113,130)(126,138)(127,137)(132,140)(133,139)Polycyclic presentation:
2^8,79|a^2=dg,b^2=de,c^2=d,d^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^78,c^b=cf,f^b=fg,g^b=gh,i^b=i^78,f^c=fg,g^c=gh1288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89727);