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Cayley diagram

C22×C4⋊AGL(1,𝔽5) #66253

order320
isoclinismisoclinic class of #700
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

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

Splittings

FittingC22×C4×C5 ⋊ C4
SylowC5 ⋊ C22×C4⋊C4

Subgroups

subgroupG/H
centerC23C2×AGL(1,𝔽5)
commutatorC2×C5C23×C4
FrattiniC2C23×AGL(1,𝔽5)
FittingC22×C4×C5C4
socleC23×C5C2×C4
prime radicalC23×D5C22
cosocleC2×D5C24
hypercenterC22×C4AGL(1,𝔽5)
hypocenterC5C22×C4⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C4

418 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:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10603801, #10603802, #10603804, …
maximal normal subgroup of#10603801, #10603802, #10603804, …
maximal quotient ofC2×D5⋊(C4⋊C8), C2×#65681, C2×#65682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1591);