Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #13691 |
200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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)(55,63)(56,68)(58,73)(59,81)(60,85)(61,76)(62,86)(64,88)(65,93)(66,95)(67,91)(69,98)(70,102)(71,99)(72,80)(74,82)(75,79)(77,90)(78,110)(83,115)(84,89)(87,106)(92,117)(94,104)(96,116)(97,101)(100,109)(103,108)(105,112)(107,113)(111,114);(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)(54,56,58,69,61,71,79,60)(55,62,64,66,67,84,77,96)(57,68,73,98,76,99,75,85)(59,70,78,100,83,103,111,82)(63,86,88,95,91,89,90,116)(65,87,92,80,94,112,101,113)(72,104,105,97,107,93,106,117)(74,81,102,110,109,115,108,114);(54,55)(56,66)(57,72)(58,77)(59,80)(60,84)(61,67)(62,69)(63,74)(64,79)(65,82)(68,97)(70,101)(71,96)(73,106)(75,105)(76,107)(78,87)(81,95)(83,113)(85,93)(86,110)(88,108)(89,114)(90,102)(91,109)(92,103)(94,100)(98,104)(99,117)(111,112)(115,116)Polycyclic presentation:
2^8,53|b^2=d,d^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ef,f^a=fg,g^a=gh,i^b=i^52,e^c=ef,f^c=fg,g^c=gh713 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90380);