Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #8501 |
616 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 616 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) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 117 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)(54,57)(55,62)(56,67)(58,73)(59,71)(60,72)(61,74)(63,79)(64,77)(65,78)(66,80)(68,84)(69,83)(70,82)(75,85)(76,81)(86,89,91,102)(87,93,94,107)(88,97,99,114)(90,103,104,101)(92,105,96,110)(95,109,108,100)(98,115,116,113)(106,112,111,117);(54,56)(55,61)(57,67)(58,70)(59,68)(60,69)(62,74)(63,76)(64,66)(65,75)(71,84)(72,83)(73,82)(77,80)(78,85)(79,81)(86,88)(87,92)(89,97)(90,98)(91,99)(93,105)(94,96)(95,106)(100,117)(101,113)(102,114)(103,115)(104,116)(107,110)(108,111)(109,112);(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)(54,55,59,64,60,65,58,63)(56,66,70,61,69,76,68,75)(57,62,71,77,72,78,73,79)(67,81,82,85,83,80,84,74)(86,87)(88,96,99,92)(89,100,102,109)(90,95)(91,94)(93,101,107,103)(97,112)(98,111,116,106)(104,108)(105,113)(110,115)(114,117)Polycyclic presentation:
2^8,53|a^2=f,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,i^a=i^52,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg,i^c=i^52710 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 61982);