Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C53×#2198 ⋊ C2 |
| Sylow | C53 ⋊ #33561 |
1096 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 137 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:
(1,5,7,19)(2,10,12,27)(3,14,4,16)(6,18,20,31)(8,21,22,24)(9,23,13,25)(11,26,28,32)(15,29,17,30)(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);(1,4,7,3)(2,9,12,13)(5,14,19,16)(6,15,20,17)(8,11,22,28)(10,25,27,23)(18,30,31,29)(21,32,24,26)(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);(1,3,7,4)(2,8,12,22)(5,16,19,14)(6,15,20,17)(9,11,13,28)(10,24,27,21)(18,30,31,29)(23,32,25,26);(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,25)(16,23)(17,22)(18,26)(19,27)(20,28)(21,29)(24,30)(31,32)Polycyclic presentation:
2^8,53|b^2=h,c^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,d^b=dh,f^b=fgh,d^c=dh,e^c=eg,i^d=i^52,f^e=fh554 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 323712);