Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C53×#359 ⋊ C2 |
| Sylow | C53 ⋊ #13021 |
462 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 462 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:
(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)(54,57)(55,64)(56,71)(58,79)(59,77)(60,86)(61,88)(62,81)(63,89)(65,94)(66,93)(67,101)(68,103)(69,96)(70,104)(72,110)(73,106)(74,111)(75,109)(76,99)(78,84)(80,83)(82,113)(85,107)(87,108)(90,112)(91,105)(92,116)(95,98)(97,117)(100,114)(102,115);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,61,87,62,75,83,74)(55,63,68,102,69,92,98,91)(57,71,88,108,81,109,80,111)(58,82,60,72,84,73,59,85)(64,89,103,115,96,116,95,105)(65,97,67,70,99,90,66,100)(76,112,93,114,94,117,101,104)(77,107,79,113,86,110,78,106);(54,55,60,67,61,68,84,99,62,69,59,66,83,98,58,65)(56,70,73,92,74,97,72,102,75,100,82,63,87,90,85,91)(57,76,79,103,80,101,77,64,81,94,78,95,88,93,86,96)(71,105,107,112,108,89,113,114,109,115,110,117,111,116,106,104)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,d^2=gh,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh,i^c=i^521190 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86685);