Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #12954 |
380 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,57,60,76)(55,62,66,93)(56,68,72,104)(58,79,81,112)(59,83,84,75)(61,86,90,100)(63,95,97,110)(64,92,65,91)(67,98,87,109)(69,106,70,107)(71,108,82,103)(73,77,111,96)(74,80,113,105)(78,102,85,101)(88,117,89,116)(94,115,99,114);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,56,60,72)(55,61)(57,68,76,104)(58,78,81,85)(59,82,84,71)(62,86)(63,94)(64,89)(65,88)(66,90)(67,77)(69,105,70,80)(73,109)(74,107,113,106)(75,108,83,103)(79,102,112,101)(87,96)(91,117)(92,116)(93,100)(95,115)(97,99)(98,111)(110,114);(54,55,59,64,60,66,84,65)(56,67,71,99,72,87,82,94)(57,73,75,95,76,111,83,110)(58,77,80,63,81,96,105,97)(61,85,88,69,90,78,89,70)(62,79,91,74,93,112,92,113)(68,100,103,117,104,86,108,116)(98,107,115,102,109,106,114,101)Polycyclic presentation:
2^8,53|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dg,e^b=egh,f^b=fh,g^b=gh,i^b=i^52,d^c=dgh,e^c=egh,f^c=fh,g^c=gh872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86291);