Finite Group Databasebeta

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C₂×#153598 #72245310

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial

Aliases

Subgroups

545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 66 points:

(1,2);(3,4,5,12,19,37,52,46,7,15,20,40,24,45,50,10,26,11,32,33,57,62,54,31,58,35,61,59,6,14,17,39,9,13,18,38,22,44,49,65,27,42,51,30,56,34,55,29,8,16,21,36,53,63,66,60,23,43,48,64,25,41,47,28);(4,10,13,29)(5,17)(6,22)(7,23)(8,26)(11,28,16,39)(12,34,38,45)(14,36,41,33)(15,46,42,65)(18,47)(19,25)(20,54)(21,55)(24,53)(27,58)(30,37,40,44)(31,35,60,43)(32,50)(48,49)(51,66)(52,61)(56,57)(59,62,64,63);(4,11,13,16)(5,18)(10,28,29,39)(12,36,38,33)(14,34,41,45)(15,35,42,43)(17,47)(20,51)(21,32)(30,59,40,64)(31,46,60,65)(37,62,44,63)(48,61)(49,52)(50,55)(54,66)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2), (3,4,5,12,19,37,52,46,7,15,20,40,24,45,50,10,26,11,32,33,57,62,54,31,58,35,61,59,6,14,17,39,9,13,18,38,22,44,49,65,27,42,51,30,56,34,55,29,8,16,21,36,53,63,66,60,23,43,48,64,25,41,47,28), (4,10,13,29)(5,17)(6,22)(7,23)(8,26)(11,28,16,39)(12,34,38,45)(14,36,41,33)(15,46,42,65)(18,47)(19,25)(20,54)(21,55)(24,53)(27,58)(30,37,40,44)(31,35,60,43)(32,50)(48,49)(51,66)(52,61)(56,57)(59,62,64,63), (4,11,13,16)(5,18)(10,28,29,39)(12,36,38,33)(14,34,41,45)(15,35,42,43)(17,47)(20,51)(21,32)(30,59,40,64)(31,46,60,65)(37,62,44,63)(48,61)(49,52)(50,55)(54,66));