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Cayley diagram

C22×C4×AGL(1,𝔽5) #66252

order320
isoclinismisoclinic class of AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC22×C4×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C2×C42)

Splittings

FittingC22×C4×C5 ⋊ C4
derived, SylowC5 ⋊ C22×C42

Subgroups

subgroupG/H
center, hypercenterC22×C4AGL(1,𝔽5)
commutator, hypocenterC5C22×C42
FrattiniC2C23×AGL(1,𝔽5)
FittingC22×C4×C5C4
socleC23×C5C2×C4
prime radicalC23×D5C22
cosocleC2×D5C24
LCSG ▷ C5
UCSC1 ◁ C22×C4

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 4 generators on 13 points:

(10,11,12,13);(5,6,7,8)(9,10,11,13,12);(3,4);(1,2)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#95, #10603800, #10603801, …
maximal normal subgroup ofAGL(1,𝔽5)×#95, #10603800, #10603801, …
maximal quotient ofC2×C4×D5⋊C8, C2×#65676, C2×C4×#700, …
automorphism group ofD5×C16, #3930, C5×#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1590);