Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #10224 |
462 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 462 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) = 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)(54,57,60,56)(55,61,64,69)(58,66,59,67)(62,68,63,65)(70,72)(71,76)(73,82)(74,79)(75,81)(77,85)(78,84)(80,83);(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)(54,56,60,57)(55,61,64,69)(58,67,59,66)(62,68,63,65)(70,71)(72,80)(73,77)(74,83)(75,78)(76,82)(79,84)(81,85);(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)(54,55,60,64)(56,65,57,68)(58,63,59,62)(61,67,69,66)(72,79)(73,74)(76,80)(77,83)(81,82)(84,85)Polycyclic presentation:
2^8,53|a^2=g,b^2=g,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,i^a=i^52,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,i^b=i^52,d^c=dh,e^c=eh,f^c=fgh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 71844);