Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #12775 |
740 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(6,13)(9,10)(12,28)(14,33)(15,16)(17,34)(19,20)(22,31)(24,25)(27,36)(29,30)(32,35)(37,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39);(5,7,11,21)(6,12,16,30)(8,23,26,18)(9,20,10,19)(13,28,15,29)(14,32,22,17)(24,36,25,27)(31,34,33,35);(1,2,3,4)(5,6,10,13,11,16,9,15)(7,17,20,34,21,32,19,35)(8,22,25,31,26,14,24,33)(12,27,28,23,30,36,29,18)(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)(86,87)(88,89)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,d^b=dh,e^b=egh,f^b=fgh,d^c=dg,e^c=eh,f^c=fh,e^d=eh,f^d=fh1022 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85448);