Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C79×#2258 ⋊ C2 |
| Sylow | C79 ⋊ #13700 |
188 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 188 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)(80,82)(81,90)(83,96)(84,103)(85,97)(86,95)(87,98)(88,106)(89,112)(91,120)(92,118)(93,117)(94,119)(99,125)(100,129)(101,124)(102,127)(104,130)(105,128)(107,131)(108,134)(109,132)(110,116)(111,133)(113,138)(114,126)(115,137)(121,142)(122,140)(123,139)(135,143)(136,141);(80,81,89,109,85,92,100,108,84,91,113,135,104,121,102,111,87,94,115,136,105,123,101,110,86,93,114,122,99,107,83,88)(82,90,112,132,97,118,129,134,103,120,138,143,130,142,127,133,98,119,137,141,128,139,124,116,95,117,126,140,125,131,96,106);(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)(81,88)(82,95)(83,89)(84,86)(85,99)(90,116)(91,110)(92,122)(93,108)(94,111)(96,126)(97,128)(98,103)(100,114)(101,113)(102,115)(104,105)(106,117)(107,109)(112,124)(118,141)(119,134)(120,133)(121,136)(123,135)(127,138)(129,137)(131,140)(132,139)(142,143)Polycyclic presentation:
2^8,79|b^2=df,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,i^a=i^78,e^c=eh3166 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90433);