Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #13035 |
456 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 115 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)(81,89)(82,96)(84,107)(85,103)(86,112)(87,106)(88,114)(90,119)(91,123)(92,102)(93,126)(94,121)(95,127)(97,131)(98,134)(99,130)(100,136)(101,133)(104,109)(105,110)(108,137)(111,132)(113,120)(115,138)(116,142)(117,141)(118,124)(122,143)(125,140)(128,139)(129,135);(80,82,86,111,87,101,110,100)(81,88,113,116,94,117,93,125)(83,96,112,132,106,133,105,136)(84,99,108,97,109,135,85,98)(89,114,120,142,121,141,126,140)(90,122,124,128,91,115,92,95)(102,127,119,143,118,139,123,138)(103,134,107,130,137,131,104,129);(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)(80,81,85,92,110,113,109,90,87,94,108,124,86,93,84,91)(82,95,99,88,111,115,97,125,101,128,135,117,100,122,98,116)(83,102,104,121,112,123,103,120,106,118,107,89,105,119,137,126)(96,114,131,139,136,142,130,138,133,141,134,127,132,140,129,143)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^78,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh1756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 86786);