Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #13536 |
318 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 97 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)(55,57)(58,60,64,75)(59,67,69,84)(61,76,62,80)(63,81,77,74)(65,73,79,72)(66,71,82,83)(68,86,78,89)(70,88,87,85);(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)(56,57)(59,66,69,82)(61,65)(62,79)(63,77)(67,83,84,71)(68,70,78,87)(72,80)(73,76)(74,81)(85,89,88,86);(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)(58,59,65,82,77,87,61,68,64,69,79,66,63,70,62,78)(60,71,80,84,81,86,72,85,75,83,76,67,74,89,73,88)Polycyclic presentation:
2^8,53|a^2=dg,b^2=deg,c^2=h,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,i^a=i^52,c^b=cf,d^b=dg,e^b=eg,g^b=gh,i^b=i^52,d^c=dgh,e^c=egh,g^c=gh,i^c=i^52866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89684);