Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #32877 |
594 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 4 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)(54,58,60,59)(55,63,64,74)(56,67,68,75)(57,71,72,73)(61,77,65,78)(62,80,69,85)(66,82,83,70)(76,79,81,84);(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)(54,57,59,73,60,72,58,71)(55,62,63,80,64,69,74,85)(56,66,67,70,68,83,75,82)(61,76,78,79,65,81,77,84);(54,56,59,75,60,68,58,67)(55,61,63,77,64,65,74,78)(57,70,73,66,72,82,71,83)(62,79,80,76,69,84,85,81);(54,55)(56,65)(57,69)(58,63)(59,74)(60,64)(61,68)(62,72)(66,81)(67,78)(70,79)(71,85)(73,80)(75,77)(76,83)(82,84)Polycyclic presentation:
2^8,53|b^2=eg,c^2=g,d^2=h,g^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=def,e^b=eh,f^b=fh,g^b=gh,d^c=de,f^c=fh,i^c=i^52860 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 318194);