Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7377 |
170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 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 — 2 generators on 117 points:
(1,3,6,17,8,19,7,18)(2,9,12,34,13,35,29,54)(4,21,23,52,24,15,30,49)(5,25,27,16,28,50,31,51)(10,14,20,45,39,47,38,46)(11,40,41,33,43,61,42,62)(22,32,56,55,26,59,36,48)(37,57,58,44,63,64,53,60)(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);(1,2,5,11,7,12,27,42,8,13,28,43,6,29,31,41)(3,14,16,44,18,46,50,64,19,47,51,60,17,45,25,57)(4,20,22,53,23,10,36,37,24,38,26,58,30,39,56,63)(9,32,33,15,34,48,40,49,35,59,62,21,54,55,61,52)(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)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^52,c^b=cg,d^b=deh,g^b=gh,d^c=dgh,e^c=eh,e^d=eh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57000);