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Cayley diagram

C2×AGL(1,𝔽9) #3842

order144
isoclinismisoclinic class of AGL(1,𝔽9)
automorphismAut(G) = C2×AΓL(1,𝔽9) (order 288) · Inn(G) = AGL(1,𝔽9) · Out(G) = C22
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽9)

Splittings

Fitting, socleC2×C32 ⋊ C8
derived, SylowC32 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽9)
commutator, hypocenterC32C2×C8
FrattiniC1G
Fitting, socleC2×C32C8
prime radicalC2×C3⋊D3C4
cosocle#157C22
LCSG ▷ C32
UCSC1 ◁ C2

27 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 18

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofGO+(4,𝔽2)⋊C4, PSU(3,𝔽2)⋊C4, AGL(1,𝔽9)⋊C4, …
maximal normal subgroup ofGO+(4,𝔽2)⋊C4, PSU(3,𝔽2)⋊C4, AGL(1,𝔽9)⋊C4, …
maximal quotient of#64213, #64214, C4×AGL(1,𝔽9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 185);