Finite Group Databasebeta

← Prev (#1070) Next (#1072) →

Cayley diagram

D53 #1071

order106
isoclinismisoclinic class of D53
automorphismAut(G) = AGL(1,𝔽53) (order 2756) · Inn(G) = G · Out(G) = C2×C13
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD53
unambiguous semidirect productC53⋊C2

Splittings

Fitting, socle, derived, SylowC53 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC53C2
LCSG ▷ C53
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 53

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

Polycyclic presentation:

2,53|b^a=b^52

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6157, C2×D53, C3×D53, …
maximal normal subgroup of#6157, C2×D53, C3×D53, …
maximal quotient ofDic53, C2×D53, C3×D53, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(106, 1);