Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #13545 |
216 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 79 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)(55,61)(56,65,66,83)(57,70)(58,71)(60,75,64,79)(62,80)(67,74,84,68)(69,85)(72,73)(76,77,82,81);(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)(55,60,61,64)(57,65)(58,74)(59,66)(62,79,69,75)(63,81,78,77)(67,72)(68,71)(70,84)(73,83)(76,85,82,80);(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)(54,55,58,63,59,61,71,78)(56,64,74,77,66,60,68,81)(57,69,72,85,73,62,70,80)(65,79,67,76,83,75,84,82)Polycyclic presentation:
2^8,53|a^2=dg,b^2=de,c^2=d,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,i^a=i^52,c^b=cf,d^b=dh,e^b=eh,f^b=fg,g^b=gh,i^b=i^52,f^c=fgh,g^c=gh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89734);