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Q8×ΓL(1,𝔽8)⋊D7 #72245174

order2352
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productQ8×ΓL(1,𝔽8)⋊D7

Splittings

FittingC72×Q8 ⋊ C2×C3
HallQ8×C7⋊D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C2×Q8
SylowC72 ⋊ C2×C3×Q8

Subgroups

subgroupG/H
center, FrattiniC2C22×ΓL(1,𝔽8)⋊D7
commutator, socle, cosocleC2×C72C23×C3
FittingC72×Q8C2×C3
prime radicalC2×ΓL(1,𝔽8)⋊D7C22
hypercenterQ8ΓL(1,𝔽8)⋊D7
hypocenterC72C2×C3×Q8
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ Q8

190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#41×ΓL(1,𝔽8)⋊D7, SD16×ΓL(1,𝔽8)⋊D7, Q8×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
maximal normal subgroup of#41×ΓL(1,𝔽8)⋊D7, SD16×ΓL(1,𝔽8)⋊D7, Q8×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.