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C3×C5×C23×AGL(1,𝔽7) #28408035

order14490
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C32×C5×C23 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×C23×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C32×C5×C23)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C3×C5×C23)
unambiguous semidirect product(C3×D7)⋊(C3×C5×C23)
unambiguous semidirect product(C5×D7)⋊(C32×C23)
unambiguous semidirect product(C3×C5×D7)⋊(C3×C23)
unambiguous semidirect product(D7×C23)⋊(C32×C5)
unambiguous semidirect product(C3×D7×C23)⋊(C3×C5)
unambiguous semidirect product(C5×D7×C23)⋊C32
unambiguous semidirect product(C3×C5×D7×C23)⋊C3

Splittings

Fitting, socleC3×C5×C7×C23 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C32×C5×C23
HallC3×C5×ΓL(1,𝔽8)×C23 ⋊ C2
HallC5×D7×C23 ⋊ C32
HallC3×ΓL(1,𝔽8)×C23 ⋊ C2×C5
HallC5×C7×C23 ⋊ C2×C32
HallD7×C23 ⋊ C32×C5
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×C23
HallC7×C23 ⋊ C2×C32×C5
HallC5×D7 ⋊ C32×C23
HallC3×ΓL(1,𝔽8) ⋊ C2×C5×C23
HallC5×C7 ⋊ C2×C32×C23
HallD7 ⋊ C32×C5×C23

Subgroups

subgroupG/H
center, hypercenterC3×C5×C23AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C2×C32×C5×C23
FrattiniC1G
Fitting, socleC3×C5×C7×C23C2×C3
LCSG ▷ C7
UCSC1 ◁ C3×C5×C23

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2415

Conjugacy classes & character table

2415 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×C5×AGL(1,𝔽7)×D23, C3×D5×C23×AGL(1,𝔽7), C3×AGL(1,𝔽7)×D115, …
maximal normal subgroup ofC3×C5×AGL(1,𝔽7)×D23, C3×D5×C23×AGL(1,𝔽7), C3×AGL(1,𝔽7)×D115, …
maximal quotient ofC3×C5×C23×ΓL(1,𝔽8)⋊C4, C2×C3×C5×C23×AGL(1,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14490, 26);