Finite Group Databasebeta

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Cayley diagram

C2×C4×C3⋊D3 #3826

order144
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = #120×AGL(2,𝔽3) (order 13824) · Inn(G) = C3⋊D3 · Out(G) = S4×#120
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC2×C4×C3⋊D3

Splittings

FittingC2×C32×C4 ⋊ C2
derived, SylowC32 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4C3⋊D3
commutator, hypocenterC32C22×C4
FrattiniC2C22×C3⋊D3
FittingC2×C32×C4C2
socleC22×C32C22
prime radicalC22×C3⋊D3C2
cosocleC2×C32C23
LCSG ▷ C32
UCSC1 ◁ C2×C4

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 72

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

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11778004, #11778005, #11778036, …
maximal subgroup of#63557, #63647, #63778, …
maximal normal subgroup of#63557, #63647, #63778, …
maximal quotient ofC4×#634, C42×C3⋊D3, #64081, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 169);