Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #29655 |
906 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 171 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 — 4 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)(57,65)(60,67)(63,69)(66,68)(70,73,75,72)(71,77,79,80)(74,82,84,81)(76,83,85,78);(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)(54,57)(55,60)(56,63)(58,65)(59,66)(61,67)(62,68)(64,69)(70,72,75,73)(71,76,79,85)(74,82,84,81)(77,83,80,78);(54,56,58,64)(55,59,61,62)(57,63,65,69)(60,66,67,68)(74,84)(76,85)(78,83)(81,82);(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)(54,55,58,61)(56,62,64,59)(57,60,65,67)(63,68,69,66)(70,71)(72,80)(73,77)(74,83)(75,79)(76,81)(78,84)(82,85)Polycyclic presentation:
2^8,53|a^2=e,b^2=e,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,i^a=i^52,c^b=cg,f^b=fh,d^c=de,f^c=fh,i^c=i^52869 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 275455);