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Cayley diagram

C3×D5×A4 #67212

order360
isoclinismisoclinic class of D5×A4
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×D5×A4
unambiguous semidirect productC5⋊(C2×C3×A4)
unambiguous semidirect product(C22×D5)⋊C32
unambiguous semidirect product(C22×C3×D5)⋊C3

Splittings

Fitting, socleC22×C3×C5 ⋊ C2×C3
derivedC22×C5 ⋊ C2×C32
HallC22×D5 ⋊ C32
SylowC5 ⋊ C2×C3×A4

Subgroups

subgroupG/H
center, hypercenterC3D5×A4
commutator, cosocle, hypocenterC22×C5C2×C32
FrattiniC1G
Fitting, socleC22×C3×C5C2×C3
LCSG ▷ C22×C5
UCSC1 ◁ C3

64 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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×A4⋊AGL(1,𝔽5), C3⋊(A4⋊AGL(1,𝔽5)), C3×A4×AGL(1,𝔽5), …
maximal normal subgroup ofC3×A4⋊AGL(1,𝔽5), C3⋊(A4⋊AGL(1,𝔽5)), C3×A4×AGL(1,𝔽5), …
maximal quotient ofC3×C5⋊(#41⋊C3), C3×D5×SL(2,𝔽3), C3×A4×Dic5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 142);