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C2×AGL(1,𝔽53) #72229201

order5512
automorphism?
homologyAbelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽53)

Splittings

Fitting, socleC2×C53 ⋊ C4×C13
derived, SylowC53 ⋊ C2×C4×C13
HallC53⋊C13 ⋊ C2×C4
HallC2×#6157 ⋊ C13

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽53)
commutator, hypocenterC53C2×C4×C13
FrattiniC1G
Fitting, socleC2×C53C4×C13
prime radicalC2×D53⋊C13C2
cosocleD53C22×C13
LCSG ▷ C53
UCSC1 ◁ C2

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 106

A minimal faithful permutation representation — 2 generators on 55 points:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽53), C22×AGL(1,𝔽53), #72275969, …
maximal normal subgroup ofC22×AGL(1,𝔽53), #72275969, C4×AGL(1,𝔽53), …
maximal quotient ofD3×AGL(1,𝔽53), C22×AGL(1,𝔽53), #72275969, …
automorphism group ofDic53, C2×D53
holomorph ofC2×C53

GAP

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

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