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Cayley diagram

C2×C3×C4×D3 #3816

order144
isoclinismisoclinic class of D3
automorphismAut(G) = C2×D3×#120 (order 384) · Inn(G) = D3 · Out(G) = C2×#120
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×C4×D3

Splittings

FittingC2×C32×C4 ⋊ C2
derivedC3 ⋊ C22×C3×C4
SylowC32 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3×C4D3
commutator, hypocenterC3C22×C3×C4
FrattiniC2C22×C3×D3
FittingC2×C32×C4C2
socleC22×C32C22
prime radicalC22×C3×D3C2
cosocleC2×C3C23×C3
LCSG ▷ C3
UCSC1 ◁ C2×C3×C4

116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D3)⋊(C3⋊C8), C3×(C2×D3)⋊C8, C2×D3×C3⋊C8, …
maximal normal subgroup of(C2×D3)⋊(C3⋊C8), C3×(C2×D3)⋊C8, C2×D3×C3⋊C8, …
maximal quotient ofC3×C4×Dic6, C3×C42×D3, C3×#872, …
automorphism group ofC22×C5×C7, Dic3×C13, C2×D3×C13, …
outer automorphism group ofC22×C5×C7, C22×C3×C13, C22×C5×C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 159);