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D8×ΓL(1,𝔽8)⋊C4 #13052046

order1344
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productD8×ΓL(1,𝔽8)⋊C4
unambiguous semidirect product(D8×Dic7)⋊C3

Splittings

HallD8×Dic7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C4×D8
SylowC7 ⋊ C3×C4×D8

Subgroups

subgroupG/H
centerC22D4×AGL(1,𝔽7)
commutatorC4×C7C22×C3×C4
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC2×C7×D8C2×C3
socleC22×C7C2×C3×D4
prime radicalC2×D8×ΓL(1,𝔽8)C2
cosocleC2×C4×C7C23×C3
hypercenterC2×D8AGL(1,𝔽7)
hypocenterC7C3×C4×D8
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

268 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD8×ΓL(1,𝔽8)⋊AGL(1,𝔽5), C4×D8×AGL(1,𝔽7), AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊C4, …
maximal normal subgroup ofC4×D8×AGL(1,𝔽7), AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊C4, #72218539, …
maximal quotient ofD8×ΓL(1,𝔽8)⋊AGL(1,𝔽5), #72276641

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 1335);