Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13008 |
516 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 516 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)(57,58)(61,77)(64,79)(66,68)(71,74)(73,75)(78,82);(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)(55,61,62,77)(57,71)(58,74)(59,76)(60,67)(63,80,70,85)(64,82,79,78)(65,83,84,81)(66,73)(68,75)(69,72);(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)(54,55,58,63,59,64,75,81,60,62,57,70,72,79,73,83)(56,65,68,78,69,80,74,77,67,84,66,82,76,85,71,61)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,d^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,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, 86604);