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Cayley diagram

C2×C11×AGL(1,𝔽5) #89192

order440
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C22×C5×AGL(1,𝔽5) (order 400) · Inn(G) = AGL(1,𝔽5) · Out(G) = C22×C5
homologyAbelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2

Aliases

direct productC2×C11×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C4×C11)

Splittings

Fitting, socleC2×C5×C11 ⋊ C4
derived, SylowC5 ⋊ C2×C4×C11
HallC5×C11 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2×C11AGL(1,𝔽5)
commutator, hypocenterC5C2×C4×C11
FrattiniC1G
Fitting, socleC2×C5×C11C4
prime radicalC2×D5×C11C2
cosocleD5C22×C11
LCSG ▷ C5
UCSC1 ◁ C2×C11

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 110

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×Dic11, (C2×AGL(1,𝔽5))⋊D11, Dic11⋊AGL(1,𝔽5), …
maximal normal subgroup ofAGL(1,𝔽5)×Dic11, (C2×AGL(1,𝔽5))⋊D11, Dic11⋊AGL(1,𝔽5), …
maximal quotient ofC11×D5⋊C8, C11×#700, C4×C11×AGL(1,𝔽5), …
automorphism group ofD5×C23, AGL(1,𝔽5)×C23

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(440, 45);