Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C79⋊#7317 |
| Sylow | C79 ⋊ #7317 |
432 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 2 generators on 111 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)(80,82,84,92,85,93,99,109)(81,86,88,101,89,90,97,106)(83,94,95,108,96,91,98,107)(87,102,103,111,104,100,105,110);(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)(80,81,83,87,85,89,96,104)(82,90,91,102,93,86,94,100)(84,97,98,103,99,88,95,105)(92,101,108,111,109,106,107,110)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,d^2=h,e^2=h,b^a=bc,c^a=cf,e^a=eg,g^a=gh,c^b=cg,d^b=df,f^b=fh,i^b=i^78,e^d=eh1276 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56869);