Finite Group Databasebeta

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C₂×#72224975 #72319888

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

Aliases

Subgroups

2234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Conjugacy classes & character table

188 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((5,15,18,36,19,37,44,39)(6,21,17,42)(7,23)(8,25)(10,26,28,51,16,35,40,53)(12,32,27,54)(13,34)(14,31)(20,47,45,38,46,63,62,64)(22,49,43,48)(24,50)(29,57,56,52,41,66,65,61)(30,58)(33,60,55,59), (3,4)(5,16,19,10)(6,12)(7,13)(8,14)(9,11)(15,35,37,26)(17,27)(18,40,44,28)(20,41,46,29)(21,32)(22,33)(23,34)(24,30)(25,31)(36,53,39,51)(38,61,64,52)(42,54)(43,55)(45,65,62,56)(47,66,63,57)(48,59)(49,60)(50,58), (1,2), (3,5,17,37,7,20,42,64,8,18,43,36,23,45,49,63,9,19,6,15,24,46,21,38,25,44,22,39,50,62,48,47)(4,10,27,35,13,29,54,61,14,28,55,51,34,56,60,66,11,16,12,26,30,41,32,52,31,40,33,53,58,65,59,57));