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#6397⋊C3 #10620451

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#6397⋊C3

Splittings

FittingC7×M16 ⋊ C2×C3
Hall#6397 ⋊ C3
HallΓL(1,𝔽8) ⋊ #100
SylowC7 ⋊ C3×#100

Subgroups

subgroupG/H
centerC2(C2×AGL(1,𝔽7))⋊C4
commutatorC22×C7C2×C3×C4
FrattiniC2×C4C2×AGL(1,𝔽7)
FittingC7×M16C2×C3
socleC2×C7C3×C22⋊C4
prime radicalC2×C4⋊AGL(1,𝔽7)C2
cosocleC2×C4×C7C22×C3
hypercenterM16AGL(1,𝔽7)
hypocenterC7C3×#100
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 56

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#100×AGL(1,𝔽7), #89521⋊C3, #89523⋊C3, …
maximal normal subgroup of#100×AGL(1,𝔽7), #89521⋊C3, #89523⋊C3, …
maximal quotient of#89259⋊C3, #89263⋊C3, #89280⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 33);