Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13607 |
332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 332 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) = 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)(57,69)(60,74)(61,78)(65,81)(68,84)(70,85)(75,76)(82,83);(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)(54,56,59,67)(55,60,63,74)(57,65,69,81)(58,72,73,66)(61,77,78,64)(62,75,71,76)(68,80,84,79)(70,82,85,83);(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)(56,64,66,79)(57,68,69,84)(58,71,73,62)(59,63)(60,65,75,82)(61,70)(67,77,72,80)(74,81,76,83)(78,85)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^52,c^b=cg,d^b=dg,e^b=efgh,f^b=fh,i^b=i^52,d^c=dh,e^d=egh,f^e=fh623 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89965);