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Cayley diagram

C4×GL(2,𝔽3) #5392

order192
isoclinismisoclinic class of #226
automorphismAut(G) = C24×S4 (order 384) · Inn(G) = S4 · Out(G) = C24
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×GL(2,𝔽3)

Splittings

FittingC4×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4S4
commutator, hypocenterSL(2,𝔽3)C2×C4
Frattini, socleC22C2×S4
FittingC4×Q8D3
prime radicalC2×GL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C22
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×GL(2,𝔽3), #73018, GL(2,𝔽3)⋊C8, …
maximal normal subgroup ofC8×GL(2,𝔽3), #73018, GL(2,𝔽3)⋊C8, …
maximal quotient ofC8×GL(2,𝔽3), #73018, C4×#859, …
automorphism group ofC32×C5, C2×C32×C5
outer automorphism group ofC32×C5, C2×C32×C5, C5×H(3,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 951);