Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #13126 |
300 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(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)(54,57,58,77,61,80,60,78)(55,63,64,93,67,95,66,75)(56,70,71,76,74,105,72,106)(59,82,69,102,84,79,83,91)(62,87,68,92,90,103,88,109)(65,97,86,108,99,94,98,113)(73,107,101,111,81,104,85,110)(89,115,100,117,96,114,112,116);(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)(54,56,69,101,58,72,84,73,61,74,83,85,60,71,59,81)(55,62,86,100,64,88,99,89,67,90,98,112,66,68,65,96)(57,76,91,107,77,70,82,110,80,106,102,104,78,105,79,111)(63,92,113,115,93,87,97,116,95,109,108,114,75,103,94,117);(54,55,59,65,60,66,83,98,61,67,84,99,58,64,69,86)(56,68,73,100,71,90,101,96,74,88,81,112,72,62,85,89)(57,75,79,108,78,95,102,97,80,93,82,113,77,63,91,94)(70,103,104,115,76,109,110,117,105,87,107,114,106,92,111,116)Polycyclic presentation:
2^8,53|a^2=f,b^2=df,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh,i^c=i^521508 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 87334);