Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C79×#1373 ⋊ C2 |
| Sylow | C79 ⋊ #7447 |
130 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 45 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 — 2 generators on 143 points:
(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)(81,89)(83,103)(84,99)(85,110)(86,113)(87,115)(88,101)(90,120)(91,125)(92,116)(93,117)(94,126)(95,119)(96,122)(97,106)(98,111)(100,109)(102,128)(104,131)(105,139)(107,141)(108,137)(112,142)(114,118)(121,129)(123,127)(124,143)(130,138)(132,140)(133,135)(134,136);(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)(80,81,84,91,109,94,111,92,88,95,85,93,87,114,86,112)(82,96,98,120,115,128,99,133,101,129,113,143,100,134,110,127)(83,102,104,124,138,122,140,135,108,136,105,90,107,121,106,123)(89,103,116,132,118,141,125,131,119,137,142,97,126,130,117,139)Polycyclic presentation:
2^8,79|a^2=d,c^2=fgh,d^2=efg,e^2=gh,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=defg,e^b=eg,f^b=fgh,g^b=gh,i^b=i^781753 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57209);