Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C53×#361 ⋊ C2 |
| Sylow | C53 ⋊ #13417 |
454 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,57,69,60,72,73,71)(55,62,63,81,66,83,61,79)(58,76,59,77,74,68,78,70)(64,82,75,85,80,84,65,67);(55,61)(56,68)(57,73)(58,59)(62,67)(63,66)(65,75)(69,77)(70,71)(72,76)(74,78)(79,82)(81,84)(83,85);(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)(56,67)(57,61)(58,75)(59,64)(60,66)(62,70)(63,73)(65,74)(68,81)(69,84)(71,82)(72,85)(76,79)(77,83)(78,80)Polycyclic presentation:
2^8,53|c^2=d,d^2=h,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fg,g^a=gh,i^a=i^52,c^b=cf,d^b=dh,e^b=eg,f^b=fgh,g^b=gh,e^c=egh,f^c=fg,g^c=gh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88970);