Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#1909 ⋊ C2 |
| Sylow | C53 ⋊ #12757 |
736 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 736 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:
(2,9)(8,22)(10,26)(11,27)(12,28)(17,25)(23,31)(24,32)(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)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84);(1,3,7,16)(2,8,11,24)(4,13,6,15)(5,19,21,14)(9,22,27,32)(10,12,17,23)(18,30,20,29)(25,31,26,28)(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);(1,2,6,10,7,11,4,17)(3,12,15,8,16,23,13,24)(5,9,20,26,21,27,18,25)(14,28,30,22,19,31,29,32)(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)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,e^b=egh,f^b=fg,i^b=i^52,d^c=dg,i^c=i^52,e^d=eh,f^d=fh788 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85358);