Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #11662 |
358 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 6784 |
A minimal faithful permutation representation — 3 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)(55,58)(59,61)(62,65,68,67)(63,70,72,83)(64,74,77,76)(66,78,82,81)(69,84,85,86)(71,79,88,89)(73,87,90,93)(75,91,92,80);(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)(62,64,68,77)(63,69,72,85)(65,76,67,74)(66,80,82,91)(70,86,83,84)(71,87,88,93)(73,89,90,79)(75,78,92,81);(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)(54,55,56,59,57,58,60,61)(62,63)(64,73)(65,70)(66,79)(67,83)(68,72)(69,75)(71,81)(74,87)(76,93)(77,90)(78,88)(80,86)(82,89)(84,91)(85,92)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,i^a=i^52,c^b=ch,d^b=df,f^b=fh,i^b=i^522138 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 79618);