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Cayley diagram

C4×C3⋊Dic3 #3749

order144
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = #120×AGL(2,𝔽3) (order 13824) · Inn(G) = C3⋊D3 · Out(G) = S4×#120
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C3×C4

Aliases

direct productC4×C3⋊Dic3

Splittings

derived, SylowC32 ⋊ C42

Subgroups

subgroupG/H
center, hypercenterC2×C4C3⋊D3
commutator, hypocenterC32C42
FrattiniC22C2×C3⋊D3
FittingC2×C32×C4C2
socle, prime radical, cosocleC22×C32C22
LCSG ▷ C32
UCSC1 ◁ C2×C4

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 14 points:

(2,3)(5,6)(11,12,13,14);(4,5,6)(7,8,9,10);(1,2,3)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11771993, #11772004, #11772014, …
maximal subgroup of#63553, #63555, #63568, …
maximal normal subgroup of#63553, #63555, #63568, …
maximal quotient ofC4×C3⋊(C3⋊C8), #63630, C8×C3⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 92);