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Cayley diagram

C11×AGL(1,𝔽5) #6354

order220
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C2×C5×AGL(1,𝔽5) (order 200) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2×C5
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)2×AGL(1,𝔽11)
homologyAbelianization (H1) = C4×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×AGL(1,𝔽5)

Splittings

Fitting, socle, HallC5×C11 ⋊ C4
derived, SylowC5 ⋊ C4×C11

Subgroups

subgroupG/H
center, hypercenterC11AGL(1,𝔽5)
commutator, hypocenterC5C4×C11
FrattiniC1G
Fitting, socleC5×C11C4
prime radicalD5×C11C2
cosocleD5C2×C11
LCSG ▷ C5
UCSC1 ◁ C11

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 55

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×D11, C2×C11×AGL(1,𝔽5), C3×C11×AGL(1,𝔽5), …
maximal normal subgroup ofAGL(1,𝔽5)×D11, C2×C11×AGL(1,𝔽5), C3×C11×AGL(1,𝔽5), …
maximal quotient ofC11×#170, C2×C11×AGL(1,𝔽5), C3×C11×AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(220, 9);