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C2×D4×ΓL(1,𝔽8)⋊C4 #13057526

order1344
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×D4×ΓL(1,𝔽8)⋊C4
unambiguous semidirect product(C2×D4×Dic7)⋊C3

Splittings

HallC2×D4×Dic7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C4×D4
SylowC7 ⋊ C2×C3×C4×D4

Subgroups

subgroupG/H
centerC23C22×AGL(1,𝔽7)
commutatorC2×C7C23×C3×C4
FrattiniC22C23×AGL(1,𝔽7)
FittingC22×C7×D4C2×C3
socleC23×C7C23×C3
prime radicalC22×D4×ΓL(1,𝔽8)C2
cosocleC22×C7C24×C3
hypercenterC22×D4AGL(1,𝔽7)
hypocenterC7C2×C3×C4×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×D4

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×ΓL(1,𝔽8)⋊AGL(1,𝔽5), C2×C4×D4×AGL(1,𝔽7), AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊C4, …
maximal normal subgroup ofC2×C4×D4×AGL(1,𝔽7), AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊C4, #120×ΓL(1,𝔽8)⋊C4, …
maximal quotient ofC2×D4×ΓL(1,𝔽8)⋊AGL(1,𝔽5), AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊C4, #120×ΓL(1,𝔽8)⋊C4

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6815);