Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #11846 |
382 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(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)(55,60)(59,67)(61,68)(62,69)(70,73,75,72)(71,77,79,81)(74,80,83,82)(76,84,85,78);(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)(54,56)(55,59)(57,65)(58,64)(60,67)(61,69)(62,68)(63,66)(70,72,75,73)(71,76,79,85)(74,82,83,80)(77,78,81,84);(54,55)(56,62,64,69)(57,61)(58,60)(59,63,67,65)(66,68)(70,71)(72,77)(73,81)(74,78,83,84)(75,79)(76,80,85,82)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=h,d^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cfh,d^b=dg,e^b=efg,d^c=dh,i^c=i^52,e^d=eh635 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80458);