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Q8×ΓL(1,𝔽8)⋊C8 #13051923

order1344
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22

Aliases

direct productQ8×ΓL(1,𝔽8)⋊C8

Splittings

HallQ8×C7⋊C8 ⋊ C3
HallΓL(1,𝔽8) ⋊ C8×Q8
SylowC7 ⋊ C3×C8×Q8

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×AGL(1,𝔽7)
commutatorC2×C7C22×C3×C8
FittingC4×C7×Q8C2×C3
socleC22×C7C22×C3×C4
prime radicalC22×ΓL(1,𝔽8)C22×C4
cosocleC2×C4×C7C23×C3
hypercenterC4×Q8AGL(1,𝔽7)
hypocenterC7C3×C8×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×Q8

204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 448

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 1212);