Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #13141 |
320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 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)(80,82,83,81)(84,87,90,101)(85,92,95,110)(86,97,98,115)(88,103,104,100)(89,105,106,114)(91,107,108,99)(93,111,102,109)(94,112,96,113);(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)(80,81,83,82)(84,86,88,89,90,98,104,106)(85,91,93,96,95,108,102,94)(87,97,103,105,101,115,100,114)(92,107,111,113,110,99,109,112);(81,82)(84,85)(86,96)(87,99,101,107)(88,102)(89,91)(90,95)(92,97,110,115)(93,104)(94,98)(100,112,103,113)(105,111,114,109)(106,108)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,c^2=dg,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,i^b=i^78,d^c=dh,e^c=egh,f^c=fh,g^c=gh1288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 87440);