Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #9406 |
554 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 554 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 103 points:
(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)(81,86)(84,87)(88,91,93,100)(89,94,95,98)(90,97,96,99)(92,102,103,101);(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)(80,81,85,87,83,86,82,84)(88,90,92,95,93,96,103,89)(91,99,102,94,100,97,101,98);(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)(81,84)(82,85)(86,87)(88,89,92,96,93,95,103,90)(91,98,101,99,100,94,102,97)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfgh,c^2=h,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^78,c^b=cg,d^b=dh,e^b=eh,f^b=fh,i^b=i^78,d^c=dh,f^c=fh1519 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 66999);