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Cayley diagram

Q8×AGL(1,𝔽5) #4134

order160
automorphismAut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = C22×AGL(1,𝔽5) · Out(G) = C2×D3
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productQ8×AGL(1,𝔽5)

Splittings

FittingC5×Q8 ⋊ C4
SylowC5 ⋊ C4×Q8

Subgroups

subgroupG/H
center, FrattiniC2C22×AGL(1,𝔽5)
commutator, socleC2×C5C22×C4
FittingC5×Q8C4
prime radical, cosocleC2×D5C23
hypercenterQ8AGL(1,𝔽5)
hypocenterC5C4×Q8
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ Q8

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSD16×AGL(1,𝔽5), #65735, Q16×AGL(1,𝔽5), …
maximal normal subgroup ofSD16×AGL(1,𝔽5), #65735, Q16×AGL(1,𝔽5), …
maximal quotient of#65703, #65707, #65709, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 209);