Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #13315 |
516 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 147 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)(81,83)(84,87)(85,93)(86,99)(88,106)(89,114)(90,107)(91,108)(92,116)(94,121)(95,126)(96,104)(97,122)(98,127)(100,131)(101,134)(102,105)(103,132)(109,135)(110,136)(111,141)(112,139)(113,140)(115,138)(117,144)(118,120)(119,143)(123,130)(124,146)(125,137)(128,147)(129,145)(133,142);(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)(84,86,91,103)(85,92,97,119)(87,105,108,134)(88,110,113,133)(89,102,90,101)(93,120,122,144)(94,123,125,128)(95,118,96,117)(98,124,129,109)(99,114,132,107)(100,111,115,112)(104,116,126,143)(106,138,140,131)(121,145,137,127)(130,146,147,135)(136,141,142,139);(80,81)(82,83)(84,85,91,97)(86,98,103,129)(87,104,108,126)(88,109,113,124)(89,96,90,95)(92,115,119,100)(93,114,122,107)(94,111,125,112)(99,130,132,147)(101,123,102,128)(105,127,134,145)(106,137,140,121)(110,117,133,118)(116,142,143,136)(120,138,144,131)(135,141,146,139)Polycyclic presentation:
2^8,79|a^2=h,b^2=h,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gh,i^b=i^78,e^c=eh,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, 88330);