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#6401⋊C3 #10620455

order672
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#6401⋊C3

Splittings

Hall#6401 ⋊ C3
HallΓL(1,𝔽8) ⋊ SD32
SylowC7 ⋊ C3×SD32

Subgroups

subgroupG/H
centerC2#1141⋊C3
commutator, cosocleC7×C8C22×C3
FrattiniC8C2×AGL(1,𝔽7)
FittingC7×D8C2×C3
socleC2×C7C3×D8
prime radicalD8×ΓL(1,𝔽8)C2
hypercenterD8AGL(1,𝔽7)
hypocenterC7C3×SD32
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89683⋊C3, #89684⋊C3, SD32×AGL(1,𝔽7), …
maximal normal subgroup of#89683⋊C3, #89684⋊C3, SD32×AGL(1,𝔽7), …
maximal quotient of#89285⋊C3, #89287⋊C3, #89357⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 37);