Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C53 ⋊ #13724 |
174 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(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)(55,59)(58,61)(62,65,66,83,86,112,90,81,69,84,87,115,93,109,78,82,70,85,88,117,107,122,110,114,92,111,89,80,68,79,67,64)(63,72,73,100,91,113,106,98,76,101,103,123,108,121,95,99,77,102,104,124,120,125,119,118,94,116,105,97,75,96,74,71);(54,56,57,60)(55,58,59,61)(62,64,78,109,68,80,87,84,92,114,90,112,107,117,66,65,70,82,67,79,93,115,89,111,69,81,110,122,86,83,88,85)(63,71,95,121,75,97,103,101,94,118,106,113,120,124,73,72,77,99,74,96,108,123,105,116,76,98,119,125,91,100,104,102);(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)(56,58)(57,59)(60,61)(62,63,70,77)(64,72,82,102)(65,71,85,99)(66,74,88,95)(67,73,78,104)(68,91,93,120)(69,94,92,76)(75,107,108,86)(79,100,109,124)(80,113,115,125)(81,116,114,101)(83,96,117,121)(84,118,111,98)(87,119,89,106)(90,105,110,103)(97,122,123,112)Polycyclic presentation:
2^8,53|a^2=h,b^2=eh,c^2=d,d^2=fg,e^2=f,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,g^a=gh,i^a=i^52,c^b=ch2126 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90559);