Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#2170 ⋊ C2 |
| Sylow | C53 ⋊ #33858 |
668 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 151 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:
(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)(54,56)(55,63)(57,72)(58,66)(59,75)(60,69)(61,78)(62,80)(64,83)(65,84)(67,71)(68,70)(73,82)(74,85)(76,77)(79,81);(54,56)(55,62)(57,68)(58,75)(59,66)(60,69)(61,77)(63,80)(64,81)(65,67)(70,72)(71,84)(73,85)(74,82)(76,78)(79,83);(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)(54,55,60,65)(56,67,69,62)(57,71,74,80)(58,64,59,76)(61,73,79,70)(63,82,84,68)(66,81,75,78)(72,83,85,77);(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)(55,61)(56,66)(57,70)(58,59)(62,63)(64,80)(65,79)(67,84)(68,82)(69,75)(71,76)(73,74)(77,78)(81,83)Polycyclic presentation:
2^8,53|b^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gh,i^a=i^52,c^b=ch,d^b=de,e^b=eg,g^b=gh,i^b=i^52,f^c=fh,e^d=eg,g^d=gh,i^d=i^521499 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 326364);