Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#1388 ⋊ C2 |
| Sylow | C53 ⋊ #13031 |
456 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 456 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)(55,63)(56,69)(58,77)(59,76)(60,86)(61,79)(62,88)(64,97)(65,92)(66,100)(67,95)(68,101)(70,108)(71,103)(72,110)(73,107)(74,98)(75,81)(78,84)(80,93)(82,112)(83,105)(85,106)(87,94)(89,114)(90,117)(91,116)(96,113)(99,115)(102,111)(104,109);(54,56,60,85,61,73,84,72)(55,62,87,90,67,91,66,99)(57,69,86,106,79,107,78,110)(58,71,81,70,82,109,59,83)(63,88,94,117,95,116,100,115)(64,96,98,102,80,89,65,68)(74,111,93,114,92,101,97,113)(75,108,112,104,76,105,77,103);(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)(54,55,59,65,84,87,82,64,61,67,81,98,60,66,58,80)(56,68,71,62,85,89,70,99,73,102,109,91,72,96,83,90)(57,74,77,95,86,97,75,94,79,92,112,63,78,93,76,100)(69,88,105,101,110,117,104,113,107,116,108,111,106,115,103,114)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,i^a=i^52,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=egh,f^c=fg,g^c=gh,i^c=i^52872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86759);