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Cayley diagram

C2×AGL(1,𝔽5) #179

order40
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C2×AGL(1,𝔽5) (order 40) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2
holomorphG ⋊ Aut(G) = #13107468
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽5)
unambiguous semidirect productD5⋊C4

Splittings

Fitting, socleC2×C5 ⋊ C4
derived, SylowC5 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽5)
commutator, hypocenterC5C2×C4
FrattiniC1G
Fitting, socleC2×C5C4
prime radicalC2×D5C2
cosocleD5C22
LCSG ▷ C5
UCSC1 ◁ C2

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 7
transitive degreeμt(G) = 10

A minimal faithful permutation representation — 2 generators on 7 points:

(4,5,6,7);(1,2)(3,4,5,7,6)

Conjugacy classes & character table

C1C2C3C4C5C6C7C8C9C10
size1455514555
order15244210244
χ11111111111
χ2111-1-1111-1-1
χ311-1i−i11-1i−i
χ411-1−ii11-1−ii
χ54-10004-1000
χ611111-1-1-1-1-1
χ7111-1-1-1-1-111
χ811-1i−i-1-11−ii
χ911-1−ii-1-11i−i
χ104-1000-41000

Here i is the imaginary unit (i² = −1).

Relationships

maximal subgroup ofC4×AGL(1,𝔽5), C4⋊AGL(1,𝔽5), C22⋊AGL(1,𝔽5), …
maximal normal subgroup ofC4×AGL(1,𝔽5), C4⋊AGL(1,𝔽5), C22⋊AGL(1,𝔽5), …
maximal quotient ofD5⋊C8, #700, C4×AGL(1,𝔽5), …
automorphism group ofDic5, C2×D5, C3×D5, …
outer automorphism group ofC3×5−1+2, C4×5−1+2
holomorph ofC2×C5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 12);