Finite Group Databasebeta

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S₄×#1580 #72342715

• solvable
• monomial

Aliases

Splittings

Subgroups

5284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

220 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((1,2,4,3), (2,3,4)(5,7,12,11,9,10,8,6)(13,15,16,17)(14,19,20,18), (14,18)(15,17)(19,20), (5,6,8,10,9,11,12,7)(13,14,16,20)(15,19,17,18));