Finite Group Databasebeta

← Prev (#5421) Next (#5423) →

Cayley diagram

C2×GU(2,𝔽3) #5422

order192
isoclinismisoclinic class of #226
automorphismAut(G) = C2×D4×S4 (order 384) · Inn(G) = S4 · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×GU(2,𝔽3)

Splittings

derivedSL(2,𝔽3) ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4S4
commutator, hypocenterSL(2,𝔽3)C2×C4
FrattiniC4C2×S4
FittingC2×#41D3
socleC22SL(2,ℤ/4)
prime radicalC2×#41⋊C3C2
cosocle#41⋊C3C22
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4

81 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 26 points:

(3,12,9,5)(6,23,16,24)(11,22,21,20)(13,25,19,26)(14,17,15,18);(1,2)(3,6,15)(4,11,14)(5,13,20)(7,18,26)(8,19,24)(9,17,23)(10,16,22)(12,21,25)

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×GU(2,𝔽3), #73067, C4⋊GU(2,𝔽3), …
maximal normal subgroup ofC4×GU(2,𝔽3), #73067, C4⋊GU(2,𝔽3), …
maximal quotient ofC2×SL(2,𝔽3)⋊C8, #73057, C4×GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 981);