Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #12026 |
180 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 180 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) = 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)(80,83,86,105,88,107,87,106)(81,90,93,102,94,119,113,136)(82,96,99,128,101,130,100,129)(84,103,109,104,110,117,85,112)(89,114,115,134,116,140,137,142)(91,118,108,135,92,121,122,138)(95,123,124,125,120,139,133,143)(97,126,98,132,131,141,111,127);(80,82,84,97,86,99,109,98,88,101,110,131,87,100,85,111)(81,89,91,95,93,115,108,124,94,116,92,120,113,137,122,133)(83,96,103,126,105,128,104,132,107,130,117,141,106,129,112,127)(90,114,118,123,102,134,135,125,119,140,121,139,136,142,138,143);(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)(80,81,85,92,87,93,110,122,88,94,109,91,86,113,84,108)(82,95,98,115,100,124,97,116,101,120,111,137,99,133,131,89)(83,102,104,135,106,119,103,121,107,136,112,138,105,90,117,118)(96,125,127,114,129,139,141,134,130,143,132,140,128,123,126,142)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,i^a=i^78,e^b=eh,e^c=eh3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 81226);