Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C4×D16×C53 ⋊ C2 |
| Sylow | C53 ⋊ #33839 |
712 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 147 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 — 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)(59,65)(60,71)(61,75)(62,76)(64,81)(67,68)(69,80)(70,88)(72,73)(74,83)(77,84)(78,79)(82,85)(87,89);(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)(60,70)(64,80)(69,81)(71,88)(72,79)(73,78)(82,87)(85,89);(58,60)(59,64)(61,71)(62,78)(63,70)(65,82)(66,85)(67,87)(68,80)(69,74)(72,75)(73,76)(77,88)(79,84)(81,83)(86,89);(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)(58,59,63,68)(60,69,70,81)(61,74,77,83)(62,65,76,67)(64,79,80,72)(66,84,86,75)(71,87,88,82)(73,89,78,85)Polycyclic presentation:
2^8,53|a^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gh,i^a=i^52,c^b=ch,d^b=de,e^b=eg,g^b=gh,i^c=i^52,e^d=eg,g^d=gh1187 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 326156);