Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #7397 |
268 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 39 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:
(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,24)(11,23)(12,25)(13,18)(14,20)(21,31)(22,32)(27,29)(28,30)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,2,5,10,7,12,6,11)(3,13,15,29,17,30,16,26)(4,18,20,27,21,28,22,9)(8,19,23,14,25,31,24,32)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)Polycyclic presentation:
2^8,53|a^2=d,c^2=fh,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,i^a=i^52,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gh,d^c=dgh,e^c=eh707 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57060);