Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #11871 |
360 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 91 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)(54,57,60,77,61,78,59,76)(55,63,65,73,66,92,85,108)(56,68,71,101,72,102,70,74)(58,81,82,112,84,75,83,111)(62,86,88,90,89,115,87,114)(64,94,79,113,96,91,95,107)(67,97,93,99,98,109,106,116)(69,103,80,110,105,100,104,117);(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)(55,62,66,89)(57,74,78,101)(58,80,84,104)(59,71,60,70)(63,90,92,114)(64,93,96,106)(65,87,85,88)(67,95,98,79)(68,77,102,76)(69,83,105,82)(73,86,108,115)(75,100,81,103)(91,97,94,109)(99,107,116,113)(110,112,117,111);(54,55,59,65,61,66,60,85)(56,67,70,93,72,98,71,106)(57,73,76,92,78,108,77,63)(58,79,82,64,84,95,83,96)(62,80,87,105,89,104,88,69)(68,99,74,109,102,116,101,97)(75,94,111,107,81,91,112,113)(86,100,114,117,115,103,90,110)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,i^b=i^52,d^c=dh,e^d=eh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80573);