Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #13270 |
316 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 107 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)(54,57,58,72,61,71,60,73)(55,63,64,69,66,75,74,56)(59,62,68,80,77,67,76,81)(65,82,79,78,83,85,84,70);(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)(54,56,68,85,58,75,77,78,61,69,76,82,60,63,59,70)(55,62,79,71,74,80,65,73,66,67,84,57,64,81,83,72);(54,55,59,65,60,64,76,79,61,66,77,83,58,74,68,84)(56,67,70,72,63,80,82,57,69,62,78,73,75,81,85,71)Polycyclic presentation:
2^8,53|a^2=f,b^2=def,c^2=e,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cg,d^b=dh,e^b=eh,i^b=i^52,d^c=dh,i^c=i^521508 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88116);