Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12437 |
214 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 59 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)(81,90)(82,97)(84,104)(85,109)(86,105)(87,110)(88,106)(89,112)(91,103)(92,122)(93,118)(94,123)(95,119)(96,124)(98,128)(99,132)(100,129)(101,133)(102,130)(107,136)(108,117)(111,137)(113,140)(114,138)(115,127)(116,139)(120,135)(121,134)(125,142)(126,141)(131,143);(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)(80,82,85,99,86,100,108,111,88,102,107,131,87,101,84,98)(81,89,92,113,93,114,121,126,95,116,120,125,94,115,91,96)(83,97,109,132,105,129,117,137,106,130,136,143,110,133,104,128)(90,112,122,140,118,138,134,141,119,139,135,142,123,127,103,124);(80,81,88,95)(82,96,102,126)(83,103,106,134)(84,92,108,120)(85,91,107,121)(86,94,87,93)(89,111,116,98)(90,117,119,104)(97,127,130,138)(99,115,131,114)(100,125,101,113)(105,135,110,122)(109,123,136,118)(112,129,139,133)(124,137,141,128)(132,142,143,140)Polycyclic presentation:
2^8,79|a^2=h,b^2=df,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,i^b=i^78,e^c=eg,g^c=gh934 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 83811);