Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C4×Q32×C53 ⋊ C2 |
| Sylow | C53 ⋊ #33853 |
696 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 6784 |
A minimal faithful permutation representation — 4 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)(58,61,63,74)(59,66,67,85)(60,70,69,75)(62,77,78,87)(64,81,82,89)(65,83,68,86)(71,73,79,84)(72,80,88,76);(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)(55,57)(58,60,63,69)(59,65,67,68)(61,70,74,75)(62,71,78,79)(64,80,82,76)(66,83,85,86)(72,81,88,89)(73,87,84,77);(58,59,63,67)(60,65,69,68)(61,72,74,88)(62,76,78,80)(64,79,82,71)(66,84,85,73)(70,81,75,89)(77,86,87,83);(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)(56,57)(59,64)(60,69)(61,70)(62,71)(65,76)(66,72)(67,82)(68,80)(74,75)(77,87)(78,79)(81,86)(83,89)(85,88)Polycyclic presentation:
2^8,53|b^2=h,c^2=h,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gh,i^a=i^52,d^b=de,e^b=eg,g^b=gh,i^c=i^52,e^d=eg,g^d=gh1199 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 326305);