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C2×#1393922 #72299263

order1024
nilpotency class2
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C214×C4

Aliases

direct productC2×#1393922

Subgroups

subgroupG/H
center, socleC25C25
commutator, Frattini, cosocleC24C26
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

78571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72297585, C2×#72297595, C2×#72237104, …
maximal normal subgroup ofC2×#72297585, C2×#72297595, C2×#72237104, …
maximal quotient ofC2×#72278211

GAP

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

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