Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7439 |
170 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 37 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)(2,9,29,34)(4,22,24,59)(5,27,28,56)(7,19,31,55)(8,20,32,18)(10,38,45,63)(11,41,46,64)(12,44,14,36)(13,35,43,51)(15,47,49,42)(16,37,52,23)(21,57,40,62)(25,54,61,53)(26,33,30,60)(39,50,58,48)(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);(1,2,5,11,8,14,7,13)(3,15,17,48,20,50,18,49)(4,21,23,58,26,45,25,47)(6,29,31,43,32,12,28,46)(9,33,34,60,36,22,44,59)(10,37,39,24,40,61,42,30)(16,51,52,41,54,64,53,35)(19,38,55,57,27,62,56,63)(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)Polycyclic presentation:
2^8,53|a^2=d,b^2=eg,c^2=fg,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=defgh,e^b=eh,g^b=gh,i^b=i^52,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, 57185);