Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13435 |
408 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 71 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)(2,9,10,35,12,36,23,56)(3,14,15,43,17,19,24,51)(6,21,25,54,29,33,59,62)(8,30,13,40,32,34,37,64)(11,18,38,48,27,49,57,53)(16,44,45,63,28,52,58,60)(31,61,41,42,39,47,46,50)(65,66)(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);(1,3,7,17)(2,8,12,32)(4,19,22,14)(5,24,26,15)(6,28,29,16)(9,34,36,30)(10,37,23,13)(11,39,27,31)(18,47,49,61)(20,43,55,51)(21,52,33,44)(25,45,59,58)(35,40,56,64)(38,41,57,46)(42,53,50,48)(54,63,62,60)(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);(1,2,7,12)(3,13,17,37)(4,18,22,49)(5,23,26,10)(6,27,29,11)(8,24,32,15)(9,33,36,21)(14,42,19,50)(16,46,28,41)(20,53,55,48)(25,38,59,57)(30,60,34,63)(31,45,39,58)(35,54,56,62)(40,52,64,44)(43,61,51,47)Polycyclic presentation:
2^8,53|a^2=h,b^2=h,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,e^b=eg,i^b=i^52,f^c=fh,g^c=gh,i^c=i^52,f^e=fh626 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89083);