Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7443 |
198 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 2 generators on 77 points:
(1,2)(3,7)(4,6)(5,8)(9,11,19,20,13,21,12,22)(10,14,15,18,17,23,16,24)(25,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27);(2,5,6,7)(9,10)(11,18,20,23,21,24,22,14)(12,15)(13,17)(16,19)(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)Polycyclic presentation:
2^8,53|a^2=d,b^2=eg,c^2=fg,d^2=gh,e^2=h,b^a=bc,c^a=ce,e^a=eg,i^a=i^52,c^b=cf,d^b=defgh,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, 57198);