Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #13020 |
460 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 119 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)(84,87,92,112)(85,94,99,128)(86,101,106,140)(88,113,90,109)(89,110,115,108)(91,119,114,111)(93,120,124,147)(95,107,97,131)(96,125,116,126)(98,133,130,127)(100,134,122,144)(102,137,104,141)(103,136,117,138)(105,142,118,139)(121,145,129,143)(123,135,132,146);(80,82,83,81)(84,86)(85,93)(87,101)(88,103)(89,104)(90,117)(91,118)(92,106)(94,120)(95,129)(96,100)(97,121)(98,132)(99,124)(102,115)(105,114)(107,143)(108,137)(109,138)(110,141)(111,142)(112,140)(113,136)(116,122)(119,139)(123,130)(125,134)(126,144)(127,135)(128,147)(131,145)(133,146);(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)(80,81,83,82)(84,85,90,97,114,130,115,96,92,99,88,95,91,98,89,116)(86,100,104,123,118,121,103,93,106,122,102,132,105,129,117,124)(87,107,110,128,119,125,113,127,112,131,108,94,111,126,109,133)(101,135,138,134,142,147,137,143,140,146,136,144,139,120,141,145)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^78,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=egh,f^c=fg,g^c=gh1762 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 86681);