Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13034 |
462 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 105 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)(54,57,61,78)(55,63,67,93)(56,69,73,108)(58,80,59,84)(60,87,81,77)(62,88,90,114)(64,95,65,99)(66,102,96,92)(68,103,104,116)(70,109,71,110)(72,111,86,107)(74,100,91,97)(75,82,76,85)(79,106,83,105)(89,115,101,113)(94,117,98,112);(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)(54,56)(55,62,67,90)(57,69)(58,79)(59,83)(60,86)(61,73)(63,88,93,114)(64,94,65,98)(66,101,96,89)(68,100,104,97)(70,82)(71,85)(72,81)(74,103,91,116)(75,110)(76,109)(77,111)(78,108)(80,106)(84,105)(87,107)(92,115,102,113)(95,117,99,112);(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)(54,55,59,64,60,66,85,97,61,67,58,65,81,96,82,100)(56,68,71,89,72,94,83,90,73,104,70,101,86,98,79,62)(57,74,76,102,77,95,80,63,78,91,75,92,87,99,84,93)(69,88,106,112,107,113,109,116,108,114,105,117,111,115,110,103)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,c^2=h,d^2=g,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^52,d^b=dg,e^b=egh,f^b=fg,g^b=gh,i^b=i^52,d^c=dgh,e^c=egh,f^c=fgh,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, 86782);