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C3×Dic5×ΓL(1,𝔽8) #11929490

order1260
isoclinismisoclinic class of D5×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×Dic5×ΓL(1,𝔽8)
unambiguous semidirect product(C7×Dic5)⋊C32
unambiguous semidirect product(C3×C7×Dic5)⋊C3

Splittings

derived, HallC5×C7 ⋊ C32×C4
HallC3×C5×ΓL(1,𝔽8) ⋊ C4
HallC7×Dic5 ⋊ C32
SylowC7 ⋊ C32×Dic5
SylowC5 ⋊ C3×C4×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2×C3D5×ΓL(1,𝔽8)
commutator, hypocenterC5×C7C32×C4
FrattiniC2C3×D5×ΓL(1,𝔽8)
Fitting, socleC2×C3×C5×C7C2×C3
prime radicalC2×C3×C5×ΓL(1,𝔽8)C2
cosocleC2×C5×C7C2×C32
LCSG ▷ C5×C7
UCSC1 ◁ C2×C3

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 420

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C11×Dic5×ΓL(1,𝔽8), C3×ΓL(1,𝔽8)×Dic55, ΓL(1,𝔽8)×Dic65⋊C3, …
maximal normal subgroup ofC3×C11×Dic5×ΓL(1,𝔽8), C3×C13×Dic5×ΓL(1,𝔽8), C3×C17×Dic5×ΓL(1,𝔽8), …
maximal quotient ofC3×C11×Dic5×ΓL(1,𝔽8), C3×ΓL(1,𝔽8)×Dic55, ΓL(1,𝔽8)×Dic65⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 42);