Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Fitting | C53×#1409 ⋊ C2 |
| Sylow | C53 ⋊ #13586 |
406 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 3 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)(54,57,59,71)(55,61,64,80)(56,66,68,69)(58,70,73,85)(60,74,76,77)(62,78,63,79)(65,81,75,83)(67,82,72,84);(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)(54,56,59,68)(55,60,64,76)(57,69,71,66)(58,72,73,67)(61,77,80,74)(62,75,63,65)(70,82,85,84)(78,81,79,83);(54,55)(56,65,67,60)(57,61)(58,63,73,62)(59,64)(66,81,82,74)(68,75,72,76)(69,83,84,77)(70,79,85,78)(71,80)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=h,d^2=gh,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dgh,e^b=efg,f^b=fh,i^c=i^52,e^d=egh,f^e=fh470 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89888);