Finite Group Databasebeta

← Prev (#6157) Next (#6159) →

Cayley diagram

C2×D53 #6158

order212
isoclinismisoclinic class of D53
automorphismAut(G) = C2×AGL(1,𝔽53) (order 5512) · Inn(G) = D53 · Out(G) = C22×C13
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD106
direct productC2×D53
unambiguous semidirect productC53⋊C22
unambiguous semidirect product(C2×C53)⋊C2

Splittings

Fitting, socleC2×C53 ⋊ C2
derived, SylowC53 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2D53
commutator, cosocle, hypocenterC53C22
FrattiniC1G
Fitting, socleC2×C53C2
LCSG ▷ C53
UCSC1 ◁ C2

10 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:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D53, D212, C22⋊D53, …
maximal normal subgroup ofC4×D53, D212, C22⋊D53, …
maximal quotient ofDic106, C4×D53, D212, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(212, 4);