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Cayley diagram

C17×AGL(1,𝔽5) #66773

order340
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C16×AGL(1,𝔽5) (order 320) · Inn(G) = AGL(1,𝔽5) · Out(G) = C16
homologyAbelianization (H1) = C4×C17 · Schur multiplier (H2) = C1

Aliases

direct productC17×AGL(1,𝔽5)

Splittings

Fitting, socle, HallC5×C17 ⋊ C4
derived, SylowC5 ⋊ C4×C17

Subgroups

subgroupG/H
center, hypercenterC17AGL(1,𝔽5)
commutator, hypocenterC5C4×C17
FrattiniC1G
Fitting, socleC5×C17C4
prime radicalD5×C17C2
cosocleD5C2×C17
LCSG ▷ C5
UCSC1 ◁ C17

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 85

A minimal faithful permutation representation — 2 generators on 22 points:

(2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(1,2,3,5,4)

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×D17, C2×C17×AGL(1,𝔽5), C3×C17×AGL(1,𝔽5), …
maximal normal subgroup ofAGL(1,𝔽5)×D17, C2×C17×AGL(1,𝔽5), C3×C17×AGL(1,𝔽5), …
maximal quotient ofC17×#170, C2×C17×AGL(1,𝔽5), C3×C17×AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(340, 7);