Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13524 |
288 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 288 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:
(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)(80,83,87,106)(81,89,94,122)(82,96,100,134)(84,108,85,112)(86,114,109,105)(88,115,117,141)(90,124,91,128)(92,121,93,120)(95,130,131,143)(97,135,136,139)(98,133,99,102)(101,126,119,125)(103,113,104,110)(107,132,111,140)(116,138,129,118)(123,137,127,142);(80,82,86,98,87,100,109,99)(81,88,93,129,94,117,92,116)(83,102,105,134,106,133,114,96)(84,107,110,97,85,111,113,136)(89,118,121,115,122,138,120,141)(90,123,126,95,91,127,125,131)(101,137,128,143,119,142,124,130)(103,132,108,139,104,140,112,135);(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)(80,81,84,90,109,92,113,125,87,94,85,91,86,93,110,126)(82,95,97,129,98,127,111,117,100,131,136,116,99,123,107,88)(83,101,103,121,114,124,112,89,106,119,104,120,105,128,108,122)(96,115,132,137,102,138,139,143,134,141,140,142,133,118,135,130)Polycyclic presentation:
2^8,79|a^2=d,b^2=de,c^2=h,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,i^a=i^78,c^b=cf,d^b=dg,e^b=egh,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh1282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89615);