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C4×ΓL(1,𝔽8)⋊D7 #11924199

order1176
isoclinismisoclinic class of ΓL(1,𝔽8)⋊D7
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×ΓL(1,𝔽8)⋊D7

Splittings

FittingC4×C72 ⋊ C2×C3
derived, SylowC72 ⋊ C2×C3×C4
HallC4×C7⋊D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4ΓL(1,𝔽8)⋊D7
commutator, hypocenterC72C2×C3×C4
FrattiniC2C2×ΓL(1,𝔽8)⋊D7
FittingC4×C72C2×C3
socle, cosocleC2×C72C22×C3
prime radicalC2×ΓL(1,𝔽8)⋊D7C2
LCSG ▷ C72
UCSC1 ◁ C4

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 84

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), D4×ΓL(1,𝔽8)⋊D7, AGL(1,𝔽5)×ΓL(1,𝔽8)⋊D7, …
maximal normal subgroup ofC4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), D4×ΓL(1,𝔽8)⋊D7, C2×C4×ΓL(1,𝔽8)⋊D7, …
maximal quotient ofAGL(1,𝔽5)×ΓL(1,𝔽8)⋊D7, C2×C4×ΓL(1,𝔽8)⋊D7, C8×ΓL(1,𝔽8)⋊D7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 114);