Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #12026 |
180 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 83 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)(54,57,60,79,62,81,61,80)(55,64,67,76,68,93,87,110)(56,70,73,102,75,104,74,103)(58,77,83,78,84,91,59,86)(63,88,89,108,90,114,111,116)(65,92,82,109,66,95,96,112)(69,97,98,99,94,113,107,117)(71,100,72,106,105,115,85,101);(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)(54,56,58,71,60,73,83,72,62,75,84,105,61,74,59,85)(55,63,65,69,67,89,82,98,68,90,66,94,87,111,96,107)(57,70,77,100,79,102,78,106,81,104,91,115,80,103,86,101)(64,88,92,97,76,108,109,99,93,114,95,113,110,116,112,117);(54,55,59,66,61,67,84,96,62,68,83,65,60,87,58,82)(56,69,72,89,74,98,71,90,75,94,85,111,73,107,105,63)(57,76,78,109,80,93,77,95,81,110,86,112,79,64,91,92)(70,99,101,88,103,113,115,108,104,117,106,114,102,97,100,116)Polycyclic presentation:
2^8,53|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,e^b=eh,i^b=i^52,e^c=eh884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 81224);