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Cayley diagram

C22×AGL(1,𝔽5) #721

order80
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = AGL(1,𝔽5)×S4 (order 480) · Inn(G) = AGL(1,𝔽5) · Out(G) = S4
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C2×C4)

Splittings

Fitting, socleC22×C5 ⋊ C4
derived, SylowC5 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC22AGL(1,𝔽5)
commutator, hypocenterC5C22×C4
FrattiniC1G
Fitting, socleC22×C5C4
prime radicalC22×D5C2
cosocleD5C23
LCSG ▷ C5
UCSC1 ◁ C22

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 3 generators on 9 points:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽5))⋊C4, C2×C4×AGL(1,𝔽5), C2×C4⋊AGL(1,𝔽5), …
maximal normal subgroup of(C2×AGL(1,𝔽5))⋊C4, C2×C4×AGL(1,𝔽5), C2×C4⋊AGL(1,𝔽5), …
maximal quotient ofC2×D5⋊C8, C2×#700, D5⋊M16, …
automorphism group of#168, C4×D5, C22⋊D5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 50);