Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #7505 |
260 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 2 generators on 89 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)(54,55,56,57)(58,60,62,72,63,70,77,88)(59,64,67,69,68,80,65,81)(61,73,74,71,75,78,76,87)(66,83,82,89,85,86,84,79);(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)(58,59,61,66,63,68,75,85)(60,69,71,86,70,81,87,83)(62,67,76,84,77,65,74,82)(64,78,79,72,80,73,89,88)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,d^b=dfg,d^c=dgh,e^d=eh,f^d=fh860 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57362);