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Cayley diagram

C22×GL(2,𝔽3) #5916

order192
isoclinismisoclinic class of #226
automorphismAut(G) = S4×#5934 (order 4608) · Inn(G) = S4 · Out(G) = #5934
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC22×GL(2,𝔽3)

Splittings

FittingC22×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C23

Subgroups

subgroupG/H
center, socle, hypercenterC23S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23
FrattiniC2C22×S4
FittingC22×Q8D3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C23

213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 32

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

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73065, C2×C4×GL(2,𝔽3), C2×C4⋊GL(2,𝔽3), …
maximal normal subgroup of#73065, C2×C4×GL(2,𝔽3), C2×C4⋊GL(2,𝔽3), …
maximal quotient ofC2×#5386, C2×C4×GL(2,𝔽3), C2×C4⋊GL(2,𝔽3), …
automorphism group ofC32×C8
outer automorphism group ofC32×C8, #3745, #3746, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1475);