Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C53 ⋊ #7534 |
128 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 25 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 — 2 generators on 85 points:
(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)(55,61,65,77)(56,66)(58,59)(62,74,76,64)(63,78,80,85)(67,82)(68,69)(70,83)(71,72)(75,81,84,79);(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)(54,55,57,63,60,65,73,80)(56,62,67,78,69,76,83,85)(58,64,72,81,59,74,71,79)(61,70,75,66,77,82,84,68)Polycyclic presentation:
2^8,53|a^2=d,b^2=g,c^2=fg,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cfg,d^b=def,e^b=egh,f^b=fh,i^b=i^52,d^c=dfg,e^d=egh,f^d=fh461 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57436);