Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13604 |
322 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 71 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 — 3 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)(54,57,60,58)(55,62,65,73)(56,67,70,71)(59,72,75,74)(61,76,78,79)(63,80,82,64)(66,77,81,83)(68,84,69,85);(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)(54,56,60,70)(55,61)(57,71,58,67)(59,68,75,69)(62,79)(63,81)(64,83)(65,78)(66,82)(72,85,74,84)(73,76)(77,80);(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)(56,66,68,79)(57,62)(58,73)(59,64,75,80)(60,65)(61,67,77,84)(63,74,82,72)(69,76,70,81)(71,83,85,78)Polycyclic presentation:
2^8,53|a^2=e,b^2=g,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,d^b=dgh,e^b=efgh,f^b=fh,i^b=i^52,e^d=egh,f^e=fh704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89950);