Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #12027 |
180 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 83 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)(80,83,86,106,88,108,87,107)(81,90,93,102,94,121,114,137)(82,96,99,130,101,103,100,131)(84,104,110,105,111,118,85,113)(89,115,116,143,117,119,140,135)(91,120,109,136,92,123,124,141)(95,125,126,127,122,142,134,138)(97,128,98,133,132,139,112,129);(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)(80,82,84,97,86,99,110,98,88,101,111,132,87,100,85,112)(81,89,91,95,93,116,109,126,94,117,92,122,114,140,124,134)(83,103,104,139,106,131,105,129,108,96,118,128,107,130,113,133)(90,119,120,142,102,135,136,138,121,115,123,125,137,143,141,127);(80,81,85,92,87,93,111,124,88,94,110,91,86,114,84,109)(82,95,98,116,100,126,97,117,101,122,112,140,99,134,132,89)(83,102,105,136,107,121,104,123,108,137,113,141,106,90,118,120)(96,127,129,115,131,142,139,143,103,138,133,119,130,125,128,135)Polycyclic presentation:
2^8,79|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,e^b=eh,i^b=i^78,e^c=eh1300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 81228);