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Cayley diagram

Q16×AGL(1,𝔽5) #65738

order320
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productQ16×AGL(1,𝔽5)

Splittings

FittingC5×Q16 ⋊ C4
SylowC5 ⋊ C4×Q16

Subgroups

subgroupG/H
centerC2D4×AGL(1,𝔽5)
commutatorC4×C5C22×C4
FrattiniC4C22×AGL(1,𝔽5)
FittingC5×Q16C4
socleC2×C5C4×D4
prime radicalC2×D5C2×D4
cosocleC4×D5C23
hypercenterQ16AGL(1,𝔽5)
hypocenterC5C4×Q16
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C4 ◁ Q16

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(2,3,4,5);(1,2,3,5,4)(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14);(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×SD32, #10604127, AGL(1,𝔽5)×Q32, …
maximal normal subgroup ofAGL(1,𝔽5)×SD32, #10604127, AGL(1,𝔽5)×Q32, …
maximal quotient of#10603968, #10603973, #10603974, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1076);