Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #13233 |
432 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(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)(80,83,86,102,87,103,85,101)(81,89,107,130,92,116,91,99)(82,95,108,134,98,126,97,125)(84,100,109,135,106,133,94,122)(88,112,137,143,114,139,113,128)(90,115,111,131,104,136,117,129)(93,118,120,140,121,141,119,123)(96,124,110,138,105,132,127,142);(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)(80,82,94,110)(81,88,111,120)(83,95,122,138)(84,105,86,108)(85,97,106,96)(87,98,109,127)(89,112,131,140)(90,93,91,113)(92,114,117,119)(99,128,115,118)(100,132,102,134)(101,125,133,124)(103,126,135,142)(104,121,107,137)(116,139,129,123)(130,143,136,141);(80,81,86,91,87,92,85,107)(82,93,97,120,98,121,108,119)(83,99,102,116,103,130,101,89)(84,104,94,117,106,90,109,111)(88,110,113,105,114,127,137,96)(95,123,125,118,126,140,134,141)(100,131,122,136,133,129,135,115)(112,124,128,138,139,132,143,142)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,c^2=eh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fg,g^b=gh,i^c=i^78,f^d=fh931 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 87925);