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Cayley diagram

C23×AGL(1,𝔽5) #4161

order160
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = AGL(1,𝔽5)×AGL(3,𝔽2) (order 26880) · Inn(G) = AGL(1,𝔽5) · Out(G) = AGL(3,𝔽2)
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC23×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C22×C4)

Splittings

Fitting, socleC23×C5 ⋊ C4
derived, SylowC5 ⋊ C23×C4

Subgroups

subgroupG/H
center, hypercenterC23AGL(1,𝔽5)
commutator, hypocenterC5C23×C4
FrattiniC1G
Fitting, socleC23×C5C4
prime radicalC23×D5C2
cosocleD5C24
LCSG ▷ C5
UCSC1 ◁ C23

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 4 generators on 11 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22⋊C4×AGL(1,𝔽5), #65699, C2×(C2×AGL(1,𝔽5))⋊C4, …
maximal normal subgroup ofC22⋊C4×AGL(1,𝔽5), #65699, C2×(C2×AGL(1,𝔽5))⋊C4, …
maximal quotient ofC22×D5⋊C8, C22×#700, C2×D5⋊M16, …
automorphism group ofC8×D5, #676, C3×#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 236);