Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13473 |
296 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 89 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)(80,83,87,106)(81,90,94,124)(82,97,100,134)(84,108,85,111)(86,114,88,105)(89,117,120,132)(91,125,92,126)(93,127,113,123)(95,128,118,139)(96,103,112,104)(98,135,110,141)(99,137,115,133)(101,107,122,109)(102,116,140,131)(119,138,129,143)(121,130,142,136);(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)(80,82,96,131,86,99,85,110,87,100,112,116,88,115,84,98)(81,89,107,130,93,119,92,118,94,120,109,136,113,129,91,95)(83,102,108,97,105,135,104,133,106,140,111,134,114,141,103,137)(90,121,125,132,123,139,101,138,124,142,126,117,127,128,122,143);(80,81,88,93,87,94,86,113)(82,95,115,130,100,118,99,136)(83,101,114,126,106,122,105,125)(84,107,112,92,85,109,96,91)(89,116,119,110,120,131,129,98)(90,108,123,103,124,111,127,104)(97,132,137,138,134,117,133,143)(102,139,141,142,140,128,135,121)Polycyclic presentation:
2^8,79|a^2=de,b^2=dg,c^2=h,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,f^b=fh,i^b=i^78,d^c=dg,e^c=eg,f^c=fg,g^c=gh1282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89310);