Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C53×#1411 ⋊ C2 |
| Sylow | C53 ⋊ #7521 |
148 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 2 generators on 85 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)(55,60)(56,66)(61,64)(62,79)(63,65)(67,85)(68,73)(69,77)(75,81)(76,84)(78,83)(80,82);(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)(54,55,57,62,58,64,70,82)(56,65,67,79,66,76,85,80)(59,63,74,83,72,84,71,81)(60,69,75,73,61,77,78,68)Polycyclic presentation:
2^8,53|a^2=d,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=ch,d^b=deg,e^b=egh,f^b=fgh,i^b=i^52,d^c=dfg,e^c=egh,f^c=fh,e^d=egh,f^d=fh464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57401);