Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #7473 |
170 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 2 generators on 147 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)(84,86,88,100,111,134,87,99,106,129,139,147,105,128,120,144,91,103,112,135,89,101,109,132,90,102,107,130,138,146,108,131)(85,92,94,115,121,133,119,127,114,141,104,125,113,124,110,126,97,118,123,143,95,116,93,98,96,117,122,142,140,145,137,136);(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)(80,81,82,83)(84,85)(86,98)(87,104)(88,110)(89,113)(90,114)(91,97)(92,99)(93,107)(94,120)(95,105)(96,106)(100,133)(101,115)(102,136)(103,127)(108,123)(109,122)(111,140)(112,137)(116,135)(117,144)(118,132)(119,139)(121,138)(124,130)(125,128)(126,129)(131,141)(134,143)(142,146)(145,147)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fg,e^2=fh,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fg,g^a=gh,i^b=i^781792 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57272);