Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7391 |
116 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 47 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:
(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)(54,56,61,73)(55,62,68,91)(57,75,78,112)(58,79,59,84)(60,86,81,72)(63,93,98,116)(64,88,83,100)(65,87,94,117)(66,90,67,89)(69,99,104,95)(70,105,110,113)(71,85,106,82)(74,103,111,102)(76,109,80,114)(77,108,92,107)(96,115,97,101);(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)(54,55,58,64,60,66,82,94,61,68,59,83,81,67,85,65)(56,69,71,93,72,102,79,101,73,104,106,116,86,103,84,115)(57,74,76,98,77,99,113,97,78,111,80,63,92,95,105,96)(62,75,87,70,89,107,100,109,91,112,117,110,90,108,88,114)Polycyclic presentation:
2^8,53|a^2=d,b^2=h,c^2=eh,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^52,c^b=ceg,d^b=dg,e^b=egh,f^b=fh,g^b=gh,d^c=dh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57042);