Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C53×#2132 ⋊ C2 |
| Sylow | C53 ⋊ #33564 |
1112 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,4,7,5)(2,10,12,18)(3,14,16,19)(6,17,22,20)(8,24,26,13)(9,27,11,28)(15,31,21,32)(23,30,25,29)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(2,9)(6,22)(8,23)(10,27)(11,12)(13,29)(15,21)(17,20)(18,28)(24,30)(25,26)(31,32)(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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,3)(2,8)(4,14)(5,19)(6,21)(7,16)(9,23)(10,24)(11,25)(12,26)(13,18)(15,22)(17,32)(20,31)(27,30)(28,29);(1,2)(3,13)(4,10)(5,18)(6,9)(7,12)(8,14)(11,22)(15,29)(16,24)(17,27)(19,26)(20,28)(21,30)(23,31)(25,32)(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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)Polycyclic presentation:
2^8,53|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,i^a=i^52,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,i^c=i^52,f^e=fh710 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 323733);