Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #7494 |
270 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 270 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:
(1,3,5,16,7,17,6,22)(2,8,11,25,12,27,21,26)(4,18,19,31,20,14,23,15)(9,24,28,32,29,13,10,30)(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);(1,2,4,10)(3,13,15,27)(5,21,19,29)(6,11,23,9)(7,12,20,28)(8,17,24,31)(14,26,22,30)(16,32,18,25)(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)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fg,g^a=gh,c^b=ch,d^b=dfh,f^b=fh,g^b=gh,i^b=i^52,d^c=dgh,f^c=fh,f^d=fh704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57333);