Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #13693 |
196 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 196 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)(55,64)(56,70)(58,75)(59,80)(60,84)(61,85)(62,77)(63,86)(65,91)(66,95)(67,98)(68,99)(69,92)(71,102)(72,104)(73,101)(74,79)(76,81)(78,109)(82,113)(83,112)(87,107)(88,115)(89,114)(90,94)(93,100)(96,117)(97,116)(103,108)(105,110)(106,111);(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)(54,56,61,72,62,73,58,60)(55,63,68,88,69,89,65,67)(57,70,85,104,77,101,75,84)(59,71,82,94,83,103,78,81)(64,86,99,115,92,114,91,98)(66,87,96,111,97,110,93,79)(74,95,107,117,106,116,105,100)(76,80,102,113,90,112,108,109);(54,55,62,69)(56,67,73,88)(57,74,77,106)(58,68,61,65)(59,79,83,111)(60,63,72,89)(64,90,92,76)(66,94,97,81)(70,100,101,117)(71,93,103,96)(75,107,85,105)(78,87,82,110)(80,98,112,115)(84,95,104,116)(86,113,114,109)(91,108,99,102)Polycyclic presentation:
2^8,53|a^2=h,b^2=dh,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, 90392);