Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #12771 |
728 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 728 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:
(1,4,8,16)(2,10,14,27)(3,6,17,24)(5,20,7,19)(9,13,28,11)(12,18,22,32)(15,21,25,23)(26,31,30,29)(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)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84);(1,3,8,17)(2,9,14,28)(4,6,16,24)(5,15,7,25)(10,13,27,11)(12,30,22,26)(18,29,32,31)(19,23,20,21)(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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,2,7,13,8,14,5,11)(3,10,15,9,17,27,25,28)(4,18,20,26,16,32,19,30)(6,22,23,29,24,12,21,31)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,c^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=egh,f^b=fg,i^b=i^52,d^c=dg,f^c=fh,i^c=i^52,e^d=eh,f^d=fh710 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85434);