Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #8495 |
620 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 107 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)(54,56,58,63)(55,60,61,69)(57,65,66,62)(59,67,64,68)(70,73)(71,77)(72,81)(74,87)(75,86)(76,89)(78,91)(79,93)(80,94)(82,99)(83,98)(84,92)(85,88)(90,96)(95,101)(97,100);(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)(56,60)(57,64)(58,61)(59,66)(62,67)(63,69)(65,68)(70,72,75,83)(71,76,78,80)(73,81,86,98)(74,82,88,100)(77,89,91,94)(79,90,92,95)(84,101,93,96)(85,97,87,99);(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)(55,59,61,64)(60,68,69,67)(70,71)(72,80,83,76)(73,84,86,93)(74,79)(75,78)(77,85,91,87)(81,96)(82,95,100,90)(88,92)(89,97)(94,99)(98,101)Polycyclic presentation:
2^8,53|a^2=f,b^2=g,c^2=h,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,i^a=i^52,d^b=dh,e^b=eh,f^b=fgh,i^b=i^52,d^c=dh,e^c=eg,f^c=fg710 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 61946);