Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #12998 |
500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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)(55,62)(57,72)(58,74)(61,76)(65,81)(67,85)(69,71)(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)(54,56,59,75,60,68,73,70)(55,61,65,83,62,76,81,78)(57,71,58,67,72,69,74,85)(63,82,64,66,79,77,80,84);(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)(54,55,58,64,59,65,72,79,60,62,74,80,73,81,57,63)(56,66,69,76,70,82,67,83,68,84,71,61,75,77,85,78)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,d^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^52,d^b=dg,e^b=eh,f^b=fg,g^b=gh,i^b=i^52,d^c=dh,f^c=fh872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86548);