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C4⋊(ΓL(1,𝔽8)⋊C8) #10620432

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

Aliases

unambiguous semidirect productC4⋊(ΓL(1,𝔽8)⋊C8)
unambiguous semidirect product(C4⋊(C7⋊C8))⋊C3

Splittings

HallC4⋊(C7⋊C8) ⋊ C3
HallΓL(1,𝔽8) ⋊ C4⋊C8
SylowC7 ⋊ C3×C4⋊C8

Subgroups

subgroupG/H
center, FrattiniC2×C4C2×AGL(1,𝔽7)
commutatorC2×C7C2×C3×C8
FittingC42×C7C2×C3
socleC22×C7C2×C3×C4
prime radicalC22×ΓL(1,𝔽8)C2×C4
cosocleC2×C4×C7C22×C3
hypercenterC42AGL(1,𝔽7)
hypocenterC7C3×C4⋊C8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C42

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89252⋊C3, #89253⋊C3, #89254⋊C3, …
maximal normal subgroup of#89252⋊C3, #89253⋊C3, #89254⋊C3, …
maximal quotient of#89252⋊C3, #89253⋊C3, C4⋊(ΓL(1,𝔽8)⋊C16), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 14);