Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#1320 ⋊ C2 |
| Sylow | C53 ⋊ #7396 |
296 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2)(3,10)(4,14)(5,8)(6,13)(7,11)(9,16)(12,15)(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)(59,60)(61,62)(63,64)(65,66)(67,68);(2,6)(3,9,12,13)(7,10)(8,16)(14,15)(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)(60,61)(62,63)(64,65)(66,67)(68,69)Polycyclic presentation:
2^8,53|a^2=d,c^2=fh,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,i^a=i^52,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gh,i^b=i^52,d^c=dgh,e^c=eh707 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57058);