Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×D16×C79 ⋊ C2 |
| Sylow | C79 ⋊ #13313 |
556 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 99 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)(81,83)(84,86)(85,92)(87,99)(88,104)(89,98)(90,100)(91,105)(93,109)(94,111)(95,97)(96,110)(101,112)(102,114)(103,113)(106,108)(107,115);(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)(85,91)(86,98)(87,101)(88,89)(92,108)(93,95)(94,106)(96,107)(97,105)(99,113)(100,104)(102,103)(109,110)(111,115);(80,81)(82,83)(84,85)(86,97)(87,91)(88,95)(89,94)(90,96)(92,98)(93,101)(99,108)(100,111)(102,106)(103,107)(104,110)(105,112)(109,113)(114,115)Polycyclic presentation:
2^8,79|d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,i^c=i^781756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 88317);