Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #12810 |
412 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 115 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)(55,62)(56,67)(58,73)(61,75)(63,78)(64,65)(66,80)(68,70)(71,79)(72,74)(76,77)(81,85);(54,56,57,68,60,67,59,70)(55,61,63,66,65,76,71,81)(58,69,72,83,73,82,74,84)(62,77,78,85,64,75,79,80);(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)(54,55,59,63,60,65,57,71)(56,66,70,76,67,81,68,61)(58,64,72,78,73,62,74,79)(69,80,83,75,82,85,84,77)Polycyclic presentation:
2^8,53|a^2=f,b^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,i^a=i^52,c^b=ch,e^c=eh,f^c=fgh,f^e=fh1178 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85605);