Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13438 |
406 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 406 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:
(1,4,5,22,8,25,7,24)(2,10,11,37,13,38,26,58)(3,15,16,45,19,47,18,21)(6,23,27,56,30,35,31,57)(9,32,14,36,34,61,39,63)(12,20,40,51,28,53,42,52)(17,46,48,62,29,59,50,55)(33,60,43,44,41,54,49,64)(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);(1,3,8,19)(2,9,13,34)(4,21,25,45)(5,18,7,16)(6,29,30,17)(10,36,38,63)(11,39,26,14)(12,41,28,33)(15,22,47,24)(20,44,53,64)(23,55,35,62)(27,48,31,50)(32,58,61,37)(40,43,42,49)(46,56,59,57)(51,60,52,54)(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);(1,2,8,13)(3,14,19,39)(4,20,25,53)(5,26,7,11)(6,28,30,12)(9,18,34,16)(10,35,38,23)(15,44,47,64)(17,49,29,43)(21,54,45,60)(22,52,24,51)(27,40,31,42)(32,55,61,62)(33,48,41,50)(36,59,63,46)(37,56,58,57)Polycyclic presentation:
2^8,53|a^2=h,b^2=h,c^2=d,d^2=h,e^2=h,f^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,f^b=fh,i^b=i^52,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, 89102);