Finite Group Databasebeta

← Prev (#6053) Next (#6055) →

Cayley diagram

D5×AGL(1,𝔽5) #6054

order200
isoclinismisoclinic class of D5×AGL(1,𝔽5)
automorphismAut(G) = AGL(1,𝔽5)2 (order 400) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productD5×AGL(1,𝔽5)
unambiguous semidirect product(C5×AGL(1,𝔽5))⋊C2
unambiguous semidirect product#1042⋊C2

Splittings

Fitting, socle, derived, SylowC52 ⋊ C2×C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC52C2×C4
prime radicalD52C2
cosocleC5×D5C22
LCSG ▷ C52
UCSC1

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 10 points:

(1,2,4,5,3)(7,8,9,10);(2,3)(4,5)(6,7,8,10,9)

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)2, C2×D5×AGL(1,𝔽5), C3×D5×AGL(1,𝔽5), …
maximal normal subgroup ofAGL(1,𝔽5)2, C2×D5×AGL(1,𝔽5), C3×D5×AGL(1,𝔽5), …
maximal quotient ofDic5×AGL(1,𝔽5), (C2×AGL(1,𝔽5))⋊D5, #87825, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 41);