Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13536 |
318 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 93 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)(84,86,90,101)(85,93,95,110)(87,102,88,106)(89,107,103,100)(91,99,105,98)(92,97,108,109)(94,112,104,115)(96,114,113,111);(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)(80,81)(82,83)(85,92,95,108)(87,91)(88,105)(89,103)(93,109,110,97)(94,96,104,113)(98,106)(99,102)(100,107)(111,115,114,112);(84,85,91,108,103,113,87,94,90,95,105,92,89,96,88,104)(86,97,106,110,107,112,98,111,101,109,102,93,100,115,99,114)Polycyclic presentation:
2^8,79|a^2=dg,b^2=deg,c^2=h,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dgh,e^c=egh,g^c=gh,i^c=i^781282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89680);