Finite Group Databasebeta

← Prev (#10620459) Next (#10620461) →

#6406⋊C3 #10620460

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect product#6406⋊C3

Splittings

Hall#6406 ⋊ C3
HallΓL(1,𝔽8) ⋊ #102
SylowC7 ⋊ C3×#102

Subgroups

subgroupG/H
centerC22(C22⋊D7)⋊C3
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C2×AGL(1,𝔽7)
FittingC2×C7×D4C2×C3
socleC22×C7C3×D4
prime radicalC2×D4×ΓL(1,𝔽8)C2
cosocleC2×C4×C7C22×C3
hypercenterC2×D4AGL(1,𝔽7)
hypocenterC7C3×#102
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89531⋊C3, #89532⋊C3, #89533⋊C3, …
maximal normal subgroup of#89531⋊C3, #89532⋊C3, #89533⋊C3, …
maximal quotient of#89324⋊C3, D4⋊(ΓL(1,𝔽8)⋊C8), #89332⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 42);