Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #32870 |
502 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 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)(54,57)(55,62)(56,67)(58,71)(59,72)(60,75)(61,77)(63,79)(64,80)(65,81)(66,82)(68,84)(69,83)(70,73)(74,78)(76,85);(55,61)(56,66)(57,70)(58,73)(60,74)(63,64)(67,83)(68,69)(71,72)(75,81)(77,79)(78,85);(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)(54,56,58,68,59,69,73,66)(55,60,61,65,64,74,63,76)(57,67,70,82,72,83,71,84)(62,78,79,81,80,75,77,85);(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)(54,55,58,63,59,64,73,61)(56,65,68,60,69,76,66,74)(57,62,71,79,72,80,70,77)(67,81,84,75,83,85,82,78)Polycyclic presentation:
2^8,53|a^2=g,b^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,i^a=i^52,c^b=cg,d^b=def,f^b=fh,i^b=i^52,d^c=de,e^c=eh,f^c=fh,g^c=gh,e^d=eh,f^d=fh704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 318155);