Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #10381 |
352 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 3392 |
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)(54,56,58,64)(55,59,61,68)(57,67,66,63)(60,69,65,62)(70,72)(71,77)(73,82)(74,80)(75,81)(76,84)(78,85)(79,83);(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)(57,66)(59,68)(60,65)(63,67)(71,76)(72,80,82,81)(77,85,84,79)(78,83);(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)(56,62)(57,65)(58,61)(59,63)(60,66)(64,69)(67,68)(70,71,73,76)(72,79)(74,78,75,83)(77,80)(81,84)(82,85)Polycyclic presentation:
2^8,53|a^2=d,b^2=dg,c^2=h,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,i^a=i^52,c^b=cf,e^b=eh,i^b=i^52,d^c=dg,e^c=egh,f^c=fg866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 72767);