Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13053 |
624 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(2,8)(5,20)(7,22)(9,27)(10,11)(12,29)(14,19)(17,21)(23,24)(25,28)(26,30)(31,32)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3,4,16,6,15,18,13)(2,7,9,26,11,24,28,12)(5,19,17,31,20,14,21,32)(8,22,27,30,10,23,25,29)(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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,2)(3,12,15,26)(4,9)(5,10,20,8)(6,11)(7,13,24,16)(14,30)(17,25,21,27)(18,28)(19,29)(22,31)(23,32)(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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)Polycyclic presentation:
2^8,53|a^2=f,b^2=df,d^2=h,e^2=h,b^a=bd,c^a=ce,e^a=egh,g^a=gh,i^a=i^52,d^b=dh,e^b=egh,f^b=fh,i^b=i^52,d^c=dg,e^c=eh,f^c=fg,e^d=eh,f^e=fh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86901);