Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #7227 |
248 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(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)(80,82,85,97,86,98,102,127,87,99,108,132,100,125,83,95)(81,88,91,111,121,131,89,94,93,113,122,140,92,112,107,124)(84,96,104,128,105,129,103,109,106,130,114,136,133,135,101,126)(90,116,118,142,141,139,115,137,120,110,134,138,119,143,117,123);(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)(80,81,84,90)(82,94,96,123)(83,91,101,118)(85,107,104,117)(86,92,105,119)(87,93,106,120)(88,109,110,127)(89,114,115,108)(95,113,126,116)(97,131,128,143)(98,111,129,138)(99,124,130,137)(100,121,133,141)(102,122,103,134)(112,136,139,132)(125,140,135,142)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fgh,e^2=fg,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,i^a=i^78,d^b=dg,d^c=dh2224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56635);