Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #11860 |
336 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(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)(54,57,58,66)(55,60,61,69)(56,63,64,65)(59,67,62,68)(70,72,75,73,76,84,74,83)(71,78,81,82,79,85,80,77);(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)(55,59)(57,65)(58,64)(60,68)(61,62)(63,66)(67,69)(70,71)(72,82)(73,78)(74,81)(75,80)(76,79)(77,84)(83,85);(54,55)(56,62,64,59)(57,60)(58,61)(63,68,65,67)(66,69)(71,77,79,82)(72,73)(74,75)(78,81,85,80)(83,84)Polycyclic presentation:
2^8,53|a^2=e,c^2=fh,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cfh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,i^c=i^52794 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80524);