Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13020 |
460 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 121 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)(58,61,66,86)(59,68,73,102)(60,75,80,114)(62,87,64,83)(63,84,89,82)(65,93,88,85)(67,94,98,121)(69,81,71,105)(70,99,90,100)(72,107,104,101)(74,108,96,118)(76,111,78,115)(77,110,91,112)(79,116,92,113)(95,119,103,117)(97,109,106,120);(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)(54,56,57,55)(58,60)(59,67)(61,75)(62,77)(63,78)(64,91)(65,92)(66,80)(68,94)(69,103)(70,74)(71,95)(72,106)(73,98)(76,89)(79,88)(81,117)(82,111)(83,112)(84,115)(85,116)(86,114)(87,110)(90,96)(93,113)(97,104)(99,108)(100,118)(101,109)(102,121)(105,119)(107,120);(54,55,57,56)(58,59,64,71,88,104,89,70,66,73,62,69,65,72,63,90)(60,74,78,97,92,95,77,67,80,96,76,106,79,103,91,98)(61,81,84,102,93,99,87,101,86,105,82,68,85,100,83,107)(75,109,112,108,116,121,111,117,114,120,110,118,113,94,115,119)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gh,i^b=i^52,d^c=dg,e^c=egh,f^c=fg,g^c=gh,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, 86680);