Finite Group Databasebeta

← Prev (#67215) Next (#67217) →

Cayley diagram

C2×C3×ΓL(1,𝔽16) #67216

order360
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphismAut(G) = C22×D3×AGL(1,𝔽5) (order 480) · Inn(G) = ΓL(1,𝔽16) · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×ΓL(1,𝔽16)
unambiguous semidirect product(C2×C3)⋊(C3×AGL(1,𝔽5))
unambiguous semidirect productD5⋊(C3×Dic3)

Splittings

Fitting, socleC2×C32×C5 ⋊ C4
derivedC3×C5 ⋊ C2×C3×C4
HallC32×C5 ⋊ C2×C4
SylowC32 ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C2×C3×Dic3

Subgroups

subgroupG/H
center, hypercenterC2×C3ΓL(1,𝔽16)
commutator, hypocenterC3×C5C2×C3×C4
FrattiniC1G
Fitting, socleC2×C32×C5C4
prime radicalC2×C32×D5C2
cosocleC3×D5C22×C3
LCSG ▷ C3×C5
UCSC1 ◁ C2×C3

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×Dic3×AGL(1,𝔽5), C3×(C2×D3)⋊AGL(1,𝔽5), C3×Dic3⋊AGL(1,𝔽5), …
maximal normal subgroup ofC3×Dic3×AGL(1,𝔽5), C3×(C2×D3)⋊AGL(1,𝔽5), C3×Dic3⋊AGL(1,𝔽5), …
maximal quotient ofC3×D5⋊(C3⋊C8), C3×#6758, C3×C4×ΓL(1,𝔽16), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 146);