Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #7407 |
240 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 39 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)(2,8)(4,21)(5,26)(6,17)(7,19)(9,37)(10,34)(11,33)(12,44)(13,35)(14,38)(15,23)(16,31)(18,22)(20,55)(24,60)(25,32)(27,52)(28,58)(29,59)(30,53)(36,50)(39,54)(40,47)(41,63)(42,64)(43,48)(45,56)(46,57)(49,62)(51,61)(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);(1,2,5,10,7,13,22,12)(3,14,16,45,19,47,17,46)(4,20,23,56,25,42,61,57)(6,28,30,41,31,11,27,43)(8,32,33,60,35,21,58,59)(9,36,38,24,39,62,40,29)(15,48,49,34,51,63,50,44)(18,54,52,55,26,37,53,64)(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,c^2=fh,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,i^a=i^52,c^b=cf,d^b=def,f^b=fh,g^b=gh,d^c=dg,e^c=eh,e^d=eh707 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57090);