Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C79×#2159 ⋊ C2 |
| Sylow | C79 ⋊ #12940 |
424 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 424 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) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(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)(85,92)(87,88)(89,102)(91,107)(93,105)(94,112)(95,111)(96,109)(97,101)(98,115)(99,106)(103,104)(108,110)(113,114);(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)(84,86)(85,91)(87,101)(88,97)(89,106)(90,100)(92,107)(93,96)(94,113)(95,110)(98,103)(99,102)(104,115)(105,109)(108,111)(112,114);(80,81,83,82)(84,85)(86,96)(87,93)(88,92)(89,105)(90,95)(91,97)(94,102)(98,109)(99,108)(100,114)(101,113)(103,112)(104,111)(106,107)(110,115)Polycyclic presentation:
2^8,79|a^2=f,b^2=f,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=egh,g^c=gh,i^c=i^781294 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 86192);