Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #12728 |
714 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 714 conjugacy classes of subgroups — too many to draw. Only its 103 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)(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)(54,57)(55,61,63,68)(56,65)(58,69)(59,70)(60,74,75,82)(62,76,71,83)(64,78,77,79)(66,80)(67,81)(72,84)(73,85);(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)(54,56,59,67)(55,60,63,75)(57,65,70,81)(58,72,73,66)(61,74,68,82)(62,77,71,64)(69,84,85,80)(76,79,83,78);(54,55)(56,64,67,77)(57,68)(58,71,73,62)(59,63)(60,66)(61,70)(65,79,81,78)(69,76,85,83)(72,75)(74,84)(80,82)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,c^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fgh,i^b=i^52,e^c=eh,i^c=i^52,f^d=fh632 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85198);