Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C53 ⋊ #7440 |
144 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2)(3,10,9,13)(4,7)(5,8)(6,12,14,16)(11,15)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(2,6,7,13,8,14,15,10)(3,9)(4,11)(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)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,c^2=fgh,d^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,i^a=i^52,c^b=cf,d^b=defg,g^b=gh,d^c=dg,e^c=eh701 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57189);