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Cayley diagram

C3×GL(2,𝔽3) #3779

order144
isoclinismisoclinic class of #226
automorphismAut(G) = C22×S4 (order 96) · Inn(G) = S4 · Out(G) = C22
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×GL(2,𝔽3)

Splittings

FittingC3×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C2×C3

Subgroups

subgroupG/H
center, socle, hypercenterC2×C3S4
commutator, cosocle, hypocenterSL(2,𝔽3)C2×C3
FrattiniC2C3×S4
FittingC3×Q8D3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3

35 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64197, #64199, #64201, …
maximal normal subgroup of#64197, #64199, #64201, …
maximal quotient ofC3×#859, C2×C3×GL(2,𝔽3), C9×GL(2,𝔽3), …
outer automorphism group of#737, C32×D7, C3⋊D21, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 122);