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C3×C5×ΓL(1,𝔽8)⋊C4 #11929489

order1260
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C32×C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×C4×C5)
unambiguous semidirect productDic7⋊(C32×C5)
unambiguous semidirect product(C3×Dic7)⋊(C3×C5)
unambiguous semidirect product(C5×Dic7)⋊C32
unambiguous semidirect product(C3×C5×Dic7)⋊C3

Splittings

derived, SylowC7 ⋊ C32×C4×C5
HallC3×C5×ΓL(1,𝔽8) ⋊ C4
HallC5×Dic7 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ C4×C5
HallC5×C7 ⋊ C32×C4
HallDic7 ⋊ C32×C5

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5AGL(1,𝔽7)
commutator, hypocenterC7C32×C4×C5
FrattiniC2C3×C5×AGL(1,𝔽7)
Fitting, socleC2×C3×C5×C7C2×C3
prime radicalC2×C3×C5×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C32×C5
LCSG ▷ C7
UCSC1 ◁ C2×C3×C5

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 420

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)⋊(Dic11⋊C5), C3×C11⋊C5×ΓL(1,𝔽8)⋊C4, C3×C5×C11×ΓL(1,𝔽8)⋊C4, …
maximal normal subgroup ofC3×C5×C11×ΓL(1,𝔽8)⋊C4, C3×C5×C13×ΓL(1,𝔽8)⋊C4, C3×C5×C17×ΓL(1,𝔽8)⋊C4, …
maximal quotient ofC3×ΓL(1,𝔽8)⋊(Dic11⋊C5), C3×C11⋊C5×ΓL(1,𝔽8)⋊C4, C3×C5×C11×ΓL(1,𝔽8)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 41);