Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C53 ⋊ #13721 |
180 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 6784 |
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)(55,59)(58,61)(62,65,66,84,89,116,111,82,69,86,90,115,96,80,67,83,70,87,91,117,122,112,93,114,95,118,110,81,68,85,79,64)(63,72,73,102,94,124,121,119,76,103,104,99,106,108,74,71,77,78,92,107,123,113,105,100,97,109,120,125,75,98,88,101);(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)(54,56,57,60)(55,58,59,61)(62,64,79,85,68,81,110,118,95,114,93,112,122,117,91,87,70,83,67,80,96,115,90,86,69,82,111,116,89,84,66,65)(63,71,88,108,75,99,120,103,97,119,105,124,123,102,92,72,77,101,74,98,106,125,104,109,76,100,121,113,94,107,73,78);(54,55)(56,58)(57,59)(60,61)(62,63)(64,78)(65,71)(66,88)(67,92)(68,94)(69,97)(70,77)(72,83)(73,79)(74,91)(75,89)(76,95)(80,102)(81,113)(82,103)(84,108)(85,107)(86,119)(87,101)(90,105)(93,104)(96,123)(98,117)(99,116)(100,118)(106,122)(109,114)(110,121)(111,120)(112,125)(115,124)Polycyclic presentation:
2^8,53|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^c=i^521058 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90543);