Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #7401 |
234 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 37 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 — 2 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)(54,56,60,71)(55,61,65,87)(57,73,77,83)(58,78,74,70)(59,69,82,68)(62,89,93,114)(63,94,90,86)(64,85,80,84)(66,91,98,92)(67,100,103,75)(72,107,109,117)(76,111,113,110)(79,104,81,106)(88,102,112,101)(95,99,96,115)(97,116,105,108);(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)(54,55,58,63,60,65,74,90)(56,66,68,97,71,98,69,105)(57,72,75,108,77,109,100,116)(59,80,81,95,82,64,79,96)(61,83,84,111,87,73,85,110)(62,88,91,76,93,112,92,113)(67,99,101,94,103,115,102,86)(70,89,104,107,78,114,106,117)Polycyclic presentation:
2^8,53|a^2=d,b^2=h,c^2=fh,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=def,f^b=fh,i^b=i^52,d^c=dg,e^d=eh701 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57071);