Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12804 |
432 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 103 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)(84,87,90,89)(85,92,95,109)(86,96,98,100)(88,102,103,101)(91,104,107,108)(93,94,110,99)(97,112,113,111)(105,114,115,106);(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)(84,86)(85,91)(87,100)(88,97)(89,96)(90,98)(92,108)(93,105)(94,106)(95,107)(99,114)(101,112)(102,111)(103,113)(104,109)(110,115);(81,83)(84,85,89,94,90,95,87,99)(86,91,96,106,98,107,100,114)(88,93,101,109,103,110,102,92)(97,105,112,104,113,115,111,108)Polycyclic presentation:
2^8,79|a^2=f,b^2=d,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch,e^c=eh,f^c=fgh,i^c=i^78,f^e=fh1282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85576);