Finite Group Databasebeta

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Cayley diagram

C4×GU(2,𝔽3) #73066

order384
isoclinismisoclinic class of #226
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C4

Aliases

direct productC4×GU(2,𝔽3)

Splittings

derivedSL(2,𝔽3) ⋊ C42

Subgroups

subgroupG/H
center, hypercenterC42S4
commutator, hypocenterSL(2,𝔽3)C42
FrattiniC2×C4C2×S4
FittingC4×#41D3
socleC22C2×SL(2,ℤ/4)
prime radical, cosocleC2×#41⋊C3C22
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C42

173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 28 points:

(5,7,10,14)(6,12)(8,16,17,25)(9,20)(11,28,13,21)(15,18,27,19)(22,26,23,24);(1,2,3,4)(5,8,16)(6,11,15)(7,13,21)(9,19,24)(10,18,27)(12,23,25)(14,22,26)(17,28,20)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×GU(2,𝔽3), #11710995, #11710996, …
maximal normal subgroup ofC8×GU(2,𝔽3), #11710995, #11710996, …
maximal quotient ofC4×SL(2,𝔽3)⋊C8, #11710988, C8×GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5642);