Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C79×#2178 ⋊ C2 |
| Sylow | C79 ⋊ #13656 |
184 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 67 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)(80,83,85,102,87,103,109,86)(81,89,91,110,94,92,93,118)(82,96,98,129,100,130,99,111)(84,101,106,134,108,135,141,107)(88,113,114,115,116,123,124,142)(90,117,120,137,122,104,121,136)(95,125,126,119,127,128,133,143)(97,105,112,138,132,140,131,139);(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)(81,88)(83,96)(84,105)(85,98)(86,111)(87,100)(89,113)(90,119)(91,114)(92,123)(93,124)(94,116)(95,104)(97,107)(99,109)(101,112)(102,129)(103,130)(106,138)(108,140)(110,115)(117,127)(118,142)(120,128)(121,125)(122,143)(126,136)(131,135)(132,134)(133,137)(139,141);(80,81,85,91,87,94,109,93)(82,95,98,126,100,127,99,133)(83,92,102,118,103,89,86,110)(84,104,106,136,108,117,141,137)(88,112,114,132,116,131,124,97)(90,101,120,134,122,135,121,107)(96,128,129,143,130,125,111,119)(105,123,138,142,140,113,139,115)Polycyclic presentation:
2^8,79|a^2=e,c^2=e,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^b=i^781768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90179);