Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #12957 |
396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2)(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)(58,60,65,76)(59,67,70,84)(61,73,79,87)(62,72,78,88)(63,75,64,74)(66,82,77,89)(68,71,80,86)(69,85,81,83);(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)(55,57)(59,66,70,77)(61,78)(62,79)(63,64)(67,82,84,89)(68,69,80,81)(71,85,86,83)(72,87)(73,88)(74,75);(54,55)(56,57)(58,59,63,69,65,70,64,81)(60,71,74,82,76,86,75,89)(61,77,62,80,79,66,78,68)(67,72,83,73,84,88,85,87)Polycyclic presentation:
2^8,53|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,d^b=dg,e^b=egh,f^b=fh,g^b=gh,i^b=i^52,d^c=dgh,e^c=eg,f^c=fh,g^c=gh,i^c=i^52872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86313);