Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #9336 |
360 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 6784 |
A minimal faithful permutation representation — 3 generators on 117 points:
(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)(54,57,59,69)(55,61,62,68)(56,65,66,80)(58,72,64,70)(60,73,75,83)(63,79,77,76)(67,82,71,81)(74,85,78,84)(86,89)(87,94)(88,98)(90,102)(91,106)(92,103)(93,107)(95,110)(96,111)(97,112)(99,114)(100,116)(101,105)(104,115)(108,117)(109,113);(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)(54,56,64,67,59,66,58,71)(55,60,63,78,62,75,77,74)(57,65,70,82,69,80,72,81)(61,73,79,84,68,83,76,85)(86,88,90,104,92,100,91,99)(87,93,95,108,96,109,105,97)(89,98,102,115,103,116,106,114)(94,107,110,117,111,113,101,112);(54,55,58,63,59,62,64,77)(56,60,67,74,66,75,71,78)(57,68,70,79,69,61,72,76)(65,73,81,84,80,83,82,85)(86,87)(88,97,100,108)(89,101)(90,95)(91,105)(92,96)(93,99,109,104)(94,102)(98,113,116,107)(103,110)(106,111)(112,114,117,115)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=egh,f^c=fg,i^c=i^52710 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 66551);