Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13362 |
670 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 670 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:
(1,4,8,24)(2,10,13,39)(3,15,18,49)(5,27,29,21)(6,23,7,22)(9,33,36,62)(11,41,25,38)(12,44,30,19)(14,45,40,64)(16,48,26,60)(17,53,31,20)(28,58,32,59)(34,37,35,61)(42,54,43,56)(46,63,52,47)(50,57,51,55)(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)(2,9)(4,20)(5,26)(6,17)(7,31)(8,18)(10,37)(11,40)(12,34)(13,36)(14,25)(15,23)(16,29)(19,33)(21,57)(22,49)(24,53)(27,55)(28,51)(30,35)(32,50)(38,47)(39,61)(41,63)(42,52)(43,46)(44,62)(45,56)(48,58)(54,64)(59,60)(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);(1,2,8,13)(3,14,18,40)(4,19,24,44)(5,25,29,11)(6,30,7,12)(9,26,36,16)(10,23,39,22)(15,47,49,63)(17,52,31,46)(20,45,53,64)(21,56,27,54)(28,42,32,43)(33,57,62,55)(34,50,35,51)(37,48,61,60)(38,58,41,59)Polycyclic presentation:
2^8,53|a^2=h,c^2=h,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,f^b=fh,g^b=gh,d^c=dg,e^c=eh,f^c=fh,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, 88624);