Finite Group Databasebeta

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C₂×#7685112 #72279834

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

Aliases

Subgroups

At least 6410 conjugacy classes of subgroups; only the 32341 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 7 generators on 22 points:

(7,8)(9,12)(10,14)(11,13)(15,16,20,22)(17,18,21,19);(3,4,5,6)(7,9)(8,12)(10,13)(11,14)(15,17,20,21)(16,18,22,19);(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,18,20,19)(16,21,22,17);(3,4,5,6)(16,22)(18,19);(3,4)(5,6)(7,8)(9,12)(10,14)(11,13)(15,17)(16,18)(19,22)(20,21);(7,10,11,12)(8,9,13,14)(15,19,20,18)(16,21,22,17);(1,2)

Conjugacy classes & character table

226 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((7,8)(9,12)(10,14)(11,13)(15,16,20,22)(17,18,21,19), (3,4,5,6)(7,9)(8,12)(10,13)(11,14)(15,17,20,21)(16,18,22,19), (3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,18,20,19)(16,21,22,17), (3,4,5,6)(16,22)(18,19), (3,4)(5,6)(7,8)(9,12)(10,14)(11,13)(15,17)(16,18)(19,22)(20,21), (7,10,11,12)(8,9,13,14)(15,19,20,18)(16,21,22,17), (1,2));