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C7⋊GU(2,𝔽3) #10620703

order672
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC7⋊GU(2,𝔽3)

Splittings

derivedC7×SL(2,𝔽3) ⋊ C4
SylowC7 ⋊ GU(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC4C7⋊S4
commutator, hypocenterC7×SL(2,𝔽3)C4
FittingC7×#41D3
socleC2×C7SL(2,ℤ/4)
prime radical, cosocleC7×#41⋊C3C2
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C4

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Polycyclic presentation:

2^2,3,7,2^3|a^2=b,b^2=g,e^2=g,f^2=g,c^a=c^2,d^a=d^6,e^a=fg,f^a=eg,e^c=f,f^c=efg,f^e=fg

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13057078, #13057080, D7×GU(2,𝔽3), …
maximal normal subgroup of#13057078, #13057080, D7×GU(2,𝔽3), …
maximal quotient ofC7⋊(SL(2,𝔽3)⋊C8), C2×C7⋊GU(2,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 285);