Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #13146 |
312 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 101 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)(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)(54,57,58,74,59,76,80,109,61,78,82,75,60,77,81,89)(55,63,64,90,83,107,111,108,66,92,85,73,65,91,79,110)(56,68,69,98,70,100,84,112,72,102,104,99,71,101,103,114)(62,86,67,93,94,115,105,117,88,113,96,106,87,97,95,116);(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)(54,56,58,69,59,70,80,84,61,72,82,104,60,71,81,103)(55,62,85,96,83,94,79,95,66,88,64,67,65,87,111,105)(57,68,74,98,76,100,109,112,78,102,75,99,77,101,89,114)(63,86,73,106,107,115,110,116,92,113,90,93,91,97,108,117);(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)(54,55,59,65,61,66,60,83)(56,67,70,95,72,96,71,105)(57,73,76,108,78,90,77,110)(58,79,80,85,82,111,81,64)(62,84,87,104,88,103,94,69)(63,89,91,74,92,109,107,75)(68,97,100,113,102,115,101,86)(93,114,116,98,106,112,117,99)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,i^a=i^52,i^b=i^52,e^c=eh,i^c=i^52884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 87480);