Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C53×#354 ⋊ C2 |
| Sylow | C53 ⋊ #33558 |
1158 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 1158 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:
(3,13)(4,15)(7,20)(8,21)(11,25)(14,28)(16,17)(22,30)(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);(1,4)(2,8)(3,12)(5,17)(6,15)(7,19)(9,22)(10,21)(11,24)(13,27)(14,29)(16,18)(20,31)(23,30)(25,32)(26,28)(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);(1,3,6,13)(2,7,10,20)(4,12,15,27)(5,14,18,28)(8,19,21,31)(9,11,23,25)(16,26,17,29)(22,24,30,32)(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);(1,2)(3,11)(4,8)(5,9)(6,10)(7,14)(12,24)(13,25)(15,21)(16,22)(17,30)(18,23)(19,26)(20,28)(27,32)(29,31)(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|b^2=h,b^a=be,c^a=cf,g^a=gh,i^a=i^52,d^b=dh,f^b=fgh,i^b=i^52,d^c=dh,e^c=eg,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, 323695);