Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C42 |
No aliases for this group.
| Sylow | C53 ⋊ #7506 |
274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 274 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 — 2 generators on 85 points:
(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)(54,56,57,68)(55,60,62,75)(58,67,71,82)(59,69,70,66)(61,78,72,74)(63,76,79,65)(64,77,80,81)(73,84,85,83);(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)(56,65,67,75)(57,62,70,80)(58,72,73,63)(59,64)(60,66,74,82)(61,71)(68,76,83,77)(69,78,84,81)(79,85)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,c^2=g,f^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,d^b=dfgh,d^c=dgh,e^d=eh,f^d=fh860 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57364);