Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #11896 |
390 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 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)(82,87)(84,86)(88,91,92,108,94,110,93,109)(89,96,99,116,100,97,98,106)(90,102,103,118,105,119,104,107)(95,111,113,117,114,101,112,115);(80,82)(81,84)(83,87)(85,86)(88,90,94,105)(89,95,100,114)(91,107,110,118)(92,104,93,103)(96,115,97,117)(98,113,99,112)(101,106,111,116)(102,108,119,109);(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)(82,86,87,84)(83,85)(88,89,92,98,94,100,93,99)(90,101,103,117,105,111,104,115)(91,106,108,97,110,116,109,96)(95,107,112,102,114,118,113,119)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,i^a=i^78,c^b=cfh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh1756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 80683);