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Cayley diagram

C3×C4×AGL(1,𝔽5) #6752

order240
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2×D4
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C4

Aliases

direct productC3×C4×AGL(1,𝔽5)

Splittings

FittingC3×C4×C5 ⋊ C4
derived, SylowC5 ⋊ C3×C42
HallC3×C5 ⋊ C42

Subgroups

subgroupG/H
center, hypercenterC3×C4AGL(1,𝔽5)
commutator, hypocenterC5C3×C42
FrattiniC2C2×C3×AGL(1,𝔽5)
FittingC3×C4×C5C4
socleC2×C3×C5C2×C4
prime radicalC2×C3×D5C22
cosocleC2×D5C22×C3
LCSG ▷ C5
UCSC1 ◁ C3×C4

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C3⋊C8, #91149, #91156, …
maximal normal subgroup ofAGL(1,𝔽5)×C3⋊C8, #91149, #91156, …
maximal quotient ofC3×C8×AGL(1,𝔽5), C3×#3992, C3×C4×#170, …
automorphism group ofD5×C13, C13×AGL(1,𝔽5)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 113);