Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #18189 |
902 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 902 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 3392 |
A minimal faithful permutation representation — 4 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)(54,57,58,64)(55,60,61,67)(56,62,63,68)(59,66,65,69)(70,73,75,74)(71,77,78,83)(72,80,82,81)(76,79,84,85);(54,56,58,63)(55,59,61,65)(57,62,64,68)(60,66,67,69);(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)(56,59)(57,60)(58,61)(62,66)(63,65)(64,67)(68,69)(70,72,75,82)(71,76,78,84)(73,81,74,80)(77,85,83,79);(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)(59,65)(66,69)(70,71,75,78)(72,79,82,85)(73,77,74,83)(76,80,84,81)Polycyclic presentation:
2^8,53|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=g,b^a=be,c^a=cf,e^a=eg,i^a=i^52,d^b=dg,e^b=eg,f^b=fh,i^b=i^52,e^c=eh872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 142432);