Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C53×#2205 ⋊ C2 |
| Sylow | C53 ⋊ #33569 |
1082 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 1082 conjugacy classes of subgroups — too many to draw. Only its 137 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 — 4 generators on 85 points:
(3,15)(8,25)(13,28)(14,19)(16,22)(24,31)(26,29)(30,32)(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);(1,4,7,5)(2,9,12,27)(3,14,15,19)(6,20,23,17)(8,24,25,31)(10,11,18,21)(13,29,28,26)(16,32,22,30)(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);(1,3)(2,8)(4,14)(5,19)(6,22)(7,15)(9,24)(10,28)(11,26)(12,25)(13,18)(16,23)(17,32)(20,30)(21,29)(27,31)(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)(3,13)(4,10)(5,18)(6,21)(7,12)(8,14)(9,17)(11,23)(15,28)(16,31)(19,25)(20,27)(22,24)(26,32)(29,30)Polycyclic presentation:
2^8,53|c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,d^b=dh,e^b=eh,f^b=fgh,i^b=i^52,e^c=eg,f^c=fh,i^c=i^52,i^d=i^52,f^e=fh554 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 323761);