Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C53 ⋊ #12698 |
990 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 990 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(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)(55,57)(58,61,63,74)(59,65,67,82)(60,69,71,72)(62,73,77,89)(64,78,79,80)(66,81,75,88)(68,84,83,86)(70,85,76,87);(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)(58,60,63,71)(59,64,67,79)(61,72,74,69)(62,76,77,70)(65,80,82,78)(66,83,75,68)(73,85,89,87)(81,84,88,86);(54,55)(56,57)(58,59)(60,68,71,83)(61,65)(62,75,77,66)(63,67)(64,70)(69,84,72,86)(73,88,89,81)(74,82)(76,79)(78,85)(80,87)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,d^b=dh,f^b=fgh,i^b=i^52,f^d=fh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85070);