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(C2×AGL(1,𝔽7))⋊C8 #10620448

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect product(C2×AGL(1,𝔽7))⋊C8
unambiguous semidirect product((C2×D7)⋊C8)⋊C3

Splittings

FittingC2×C7×C8 ⋊ C2×C3
Hall(C2×D7)⋊C8 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22⋊C8
SylowC7 ⋊ C3×C22⋊C8

Subgroups

subgroupG/H
center, FrattiniC2×C4C2×AGL(1,𝔽7)
commutatorC2×C7C2×C3×C8
FittingC2×C7×C8C2×C3
socleC22×C7C2×C3×C4
prime radicalC22×AGL(1,𝔽7)C4
cosocleC2×C4×C7C22×C3
hypercenterC2×C8AGL(1,𝔽7)
hypocenterC7C3×C22⋊C8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C8

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

2^2,3,2^3,7|b^2=d,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#89457⋊C3, C8×C4⋊AGL(1,𝔽7), #89460⋊C3, …
maximal normal subgroup of#89457⋊C3, C8×C4⋊AGL(1,𝔽7), #89460⋊C3, …
maximal quotient of#89254⋊C3, #89251⋊C3, #89263⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 30);