Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13431 |
402 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,4,5,20,7,22,26,55)(2,9,10,34,12,35,23,57)(3,14,15,44,17,19,24,52)(6,28,25,59,29,21,56,54)(8,30,13,41,32,33,36,64)(11,39,37,53,40,18,49,50)(16,47,46,61,27,45,58,63)(31,48,42,51,38,62,60,43)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117);(1,3,7,17)(2,8,12,32)(4,19,22,14)(5,24,26,15)(6,27,29,16)(9,33,35,30)(10,36,23,13)(11,38,40,31)(18,48,39,62)(20,44,55,52)(21,47,28,45)(25,46,56,58)(34,41,57,64)(37,42,49,60)(43,53,51,50)(54,63,59,61)(65,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67);(1,2,7,12)(3,13,17,36)(4,18,22,39)(5,23,26,10)(6,11,29,40)(8,24,32,15)(9,28,35,21)(14,43,19,51)(16,42,27,60)(20,53,55,50)(25,49,56,37)(30,61,33,63)(31,58,38,46)(34,54,57,59)(41,47,64,45)(44,62,52,48)Polycyclic presentation:
2^8,53|a^2=h,b^2=h,c^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,c^b=cf,d^b=dh,e^b=eg,e^c=eh,f^c=fh,g^c=gh,i^c=i^52,f^e=fh704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89055);