Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13356 |
676 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(1,4,7,21)(2,9,12,34)(3,14,17,44)(5,24,26,19)(6,28,29,20)(8,30,32,60)(10,36,22,33)(11,38,27,18)(13,39,35,62)(15,42,23,52)(16,47,48,43)(25,53,57,56)(31,61,58,49)(37,50,54,51)(40,63,46,41)(45,59,55,64)(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);(1,3)(2,8)(4,14)(5,23)(6,16)(7,17)(9,30)(10,35)(11,31)(12,32)(13,22)(15,26)(18,49)(19,52)(20,43)(21,44)(24,42)(25,55)(27,58)(28,47)(29,48)(33,62)(34,60)(36,39)(37,46)(38,61)(40,54)(41,51)(45,57)(50,63)(53,59)(56,64)(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)(3,13,17,35)(4,18,21,38)(5,22,26,10)(6,27,29,11)(8,23,32,15)(9,28,34,20)(14,41,44,63)(16,46,48,40)(19,51,24,50)(25,37,57,54)(30,59,60,64)(31,45,58,55)(33,53,36,56)(39,47,62,43)(42,49,52,61)Polycyclic presentation:
2^8,53|a^2=h,c^2=h,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,g^b=gh,d^c=dg,e^c=eh,i^c=i^52,f^d=fh782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88588);