Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#1536 ⋊ C2 |
| Sylow | C53 ⋊ #13566 |
478 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 478 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:
(1,4,7,6)(2,9,12,21)(3,14,18,17)(5,19,20,22)(8,23,24,25)(10,26,27,11)(13,28,30,29)(15,16,31,32)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3)(2,8)(4,17)(5,15)(6,14)(7,18)(9,25)(10,13)(11,28)(12,24)(16,22)(19,32)(20,31)(21,23)(26,29)(27,30)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,2,5,11,7,12,20,26)(3,13,16,24,18,30,32,8)(4,9,19,10,6,21,22,27)(14,28,31,25,17,29,15,23)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)Polycyclic presentation:
2^8,53|a^2=e,c^2=h,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,e^b=efh,f^b=fh,g^b=gh,i^b=i^52,e^d=egh,f^d=fh,f^e=fh704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89812);