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C4⋊(C3⋊(C7⋊AGL(1,𝔽5))) #13132627

order1680
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC4⋊(C3⋊(C7⋊AGL(1,𝔽5)))

Splittings

FittingC3×C4×C5×C7 ⋊ C4
HallC3×C5×C7 ⋊ C4⋊C4
HallC5×C7 ⋊ C4⋊Dic3
HallC3×C7 ⋊ C4⋊AGL(1,𝔽5)
HallC3×C5 ⋊ C4⋊Dic7
SylowC7 ⋊ C4⋊ΓL(1,𝔽16)
SylowC5 ⋊ C4⋊Dic21
SylowC3 ⋊ C4⋊(C7⋊AGL(1,𝔽5))

Subgroups

subgroupG/H
center, FrattiniC2C2×C3⋊(C7⋊AGL(1,𝔽5))
commutator, socleC2×C3×C5×C7C2×C4
FittingC3×C4×C5×C7C4
prime radical, cosocleC2×C3×C7×D5C22
hypercenterC4C3⋊(C7⋊AGL(1,𝔽5))
hypocenterC3×C5×C7C4⋊C4
LCSG ▷ C2×C3×C5×C7 ▷ C3×C5×C7
UCSC1 ◁ C2 ◁ C4

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 420

Polycyclic presentation:

2^4,3,5,7|a^2=c,b^2=d,b^a=bd,e^a=e^2,f^a=f^2,g^a=g^6,f^c=f^4

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 647);