Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #10297 |
528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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 — 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)(57,67)(59,68)(60,65)(63,66)(70,72,74,80)(71,75,77,84)(73,83,82,79)(76,85,81,78);(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)(54,56,58,64)(55,59,61,68)(57,66,67,63)(60,69,65,62)(73,82)(75,84)(76,81)(79,83);(54,55,58,61)(56,62,64,69)(57,65,67,60)(59,66,68,63)(70,71)(72,78)(73,81)(74,77)(75,79)(76,82)(80,85)(83,84)Polycyclic presentation:
2^8,53|a^2=g,b^2=g,c^2=h,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,d^c=dg,e^c=eh,i^c=i^52872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 72302);