Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #13708 |
158 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 125 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)(62,64,65,75)(63,70,69,91)(66,83,79,110)(67,88,80,113)(68,90,81,78)(71,98,92,122)(72,101,93,123)(73,102,89,96)(74,104,103,82)(76,97,94,85)(77,100,106,84)(86,120,111,117)(87,95,112,121)(99,125,114,108)(105,115,109,119)(107,118,116,124);(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)(54,56,57,61)(55,58,59,60)(63,69)(70,91)(71,92)(72,93)(73,89)(74,103)(82,104)(87,112)(95,121)(96,102)(98,122)(99,114)(101,123)(105,109)(108,125)(115,119);(54,55,57,59)(56,60,61,58)(62,63,65,69)(64,74,75,103)(66,82,79,104)(67,87,80,112)(68,89,81,73)(70,94,91,76)(71,97,92,85)(72,100,93,84)(77,108,106,125)(78,109,90,105)(83,101,110,123)(86,119,111,115)(88,98,113,122)(95,120,121,117)(96,107,102,116)(99,118,114,124)Polycyclic presentation:
2^8,53|a^2=d,b^2=dh,c^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh,i^c=i^521037 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90474);