Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#1277 ⋊ C2 |
| Sylow | C53 ⋊ #10278 |
562 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 562 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 81 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)(58,60,63,59)(61,65,62,64)(67,73)(68,77)(69,70)(74,75)(76,80)(78,79);(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)(59,64)(60,65)(61,62)(66,68)(67,72)(69,78)(70,77)(71,79)(73,80)(74,76)(75,81);(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)(58,59,63,60)(61,64,62,65)(66,67)(68,76)(69,74)(70,73)(71,75)(72,77)(78,81)(79,80)Polycyclic presentation:
2^8,53|a^2=gh,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,i^a=i^52,c^b=cf,d^b=dg,f^b=fg,i^b=i^52,d^c=dg,e^c=eh,f^c=fg,i^c=i^521502 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 72189);