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#6411⋊C3 #10620465

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

Aliases

unambiguous semidirect product#6411⋊C3

Splittings

Hall#6411 ⋊ C3
HallΓL(1,𝔽8) ⋊ #104
SylowC7 ⋊ C3×#104

Subgroups

subgroupG/H
centerC4(C22⋊D7)⋊C3
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C2×AGL(1,𝔽7)
FittingC7×#41C2×C3
socleC2×C7C3×C22⋊C4
prime radical#41×ΓL(1,𝔽8)C2
cosocleC2×C4×C7C22×C3
hypercenter#41AGL(1,𝔽7)
hypocenterC7C3×#104
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 56

Polycyclic presentation:

2^2,3,2^3,7|a^2=d,d^2=f,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#104×AGL(1,𝔽7), #89593⋊C3, #89916⋊C3, …
maximal normal subgroup of#104×AGL(1,𝔽7), #89593⋊C3, #89916⋊C3, …
maximal quotient of#89327⋊C3, D4⋊(ΓL(1,𝔽8)⋊C8), (Q8⋊(C7⋊C8))⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 47);