Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7419 |
152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 2 generators on 93 points:
(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)(55,58)(57,61)(59,60)(62,63,68,70)(64,75,79,86)(65,81,66,83)(67,84,82,73)(69,74,85,77)(71,76,72,80)(78,88,91,89)(87,93,90,92);(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)(54,55)(56,60,61,58)(57,59)(63,69,70,85)(64,74)(65,80)(66,76)(67,82)(71,88,72,89)(73,90,84,87)(75,81,86,83)(77,79)(78,93)(91,92)Polycyclic presentation:
2^8,53|a^2=d,b^2=h,c^2=fg,e^2=gh,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,i^a=i^52,c^b=cf,d^b=defg,e^b=eg,f^b=fg,g^b=gh1037 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57126);