Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #9416 |
380 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 101 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)(54,57)(55,61)(56,65)(58,64)(59,60)(62,67)(63,66)(68,69)(70,73,75,85)(71,77,78,92)(72,80,81,95)(74,87,88,84)(76,89,91,100)(79,93,86,83)(82,99,97,96)(90,101,98,94);(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)(54,56,58,61,59,67,69,66)(55,60,62,68,63,57,65,64)(70,72)(71,76,78,91)(73,80)(74,82)(75,81)(77,89,92,100)(79,90,86,98)(83,94,93,101)(84,96)(85,95)(87,99)(88,97);(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)(56,64,67,68)(57,61,60,66)(58,62)(59,63)(65,69)(70,71)(72,76,81,91)(73,83)(74,86)(75,78)(77,84)(79,88)(80,94,95,101)(82,98,97,90)(85,93)(87,92)(89,99,100,96)Polycyclic presentation:
2^8,53|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^52,d^b=dh,e^b=egh,f^b=fh,i^b=i^52,d^c=dgh,e^c=eg,f^c=fgh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 67052);