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 ⋊ #11749 |
330 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 93 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)(54,56,58,57)(55,59,60,61);(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)(54,55)(56,59)(57,61)(58,60)(62,64)(63,69)(65,78)(66,75)(67,83)(68,76)(70,87)(71,84)(72,86)(73,77)(74,80)(79,91)(81,93)(82,85)(88,89)(90,92);(55,59,60,61)(62,63,66,71,68,72,67,82)(64,73,75,89,76,90,83,87)(65,77,79,88,81,92,80,70)(69,78,84,91,86,93,85,74)Polycyclic presentation:
2^8,53|a^2=e,c^2=fgh,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,i^c=i^52812 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80003);