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#6372⋊C3 #10620426

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#6372⋊C3

Splittings

FittingC7×C16 ⋊ C2×C3
Hall#6372 ⋊ C3
HallΓL(1,𝔽8) ⋊ M32
SylowC7 ⋊ C3×M32

Subgroups

subgroupG/H
center, FrattiniC8C2×AGL(1,𝔽7)
commutator, socleC2×C7C2×C3×C8
FittingC7×C16C2×C3
prime radicalC2×AGL(1,𝔽7)C8
cosocleC7×C8C22×C3
hypercenterC16AGL(1,𝔽7)
hypocenterC7C3×M32
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C8 ◁ C16

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6372⋊C3, #89673⋊C3, M32×AGL(1,𝔽7), …
maximal normal subgroup ofC2×#6372⋊C3, #89673⋊C3, M32×AGL(1,𝔽7), …
maximal quotient of(ΓL(1,𝔽8)⋊C4)⋊C16, #89297⋊C3, (C2×AGL(1,𝔽7))⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 8);