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C2×C7×ΓL(1,𝔽8)×C11⋊C5 #30633512

order16170
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC2×C7×ΓL(1,𝔽8)×C11⋊C5
unambiguous semidirect product(C2×C7)⋊(C3×C7×C11⋊C5)
unambiguous semidirect product(C2×C11)⋊(C5×C7×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C11×ΓL(1,𝔽8))⋊(C5×C7)
unambiguous semidirect product(C2×C7×C11⋊C5)⋊(C3×C7)
unambiguous semidirect product(C2×C7×C11×ΓL(1,𝔽8))⋊C5

Splittings

Fitting, socle, HallC2×C72×C11 ⋊ C3×C5
derivedC7×C11 ⋊ C2×C3×C5×C7
HallC2×C72×C11⋊C5 ⋊ C3
HallC2×C7×C11×ΓL(1,𝔽8) ⋊ C5
HallC72×C11⋊C5 ⋊ C2×C3
HallC7×C11×ΓL(1,𝔽8) ⋊ C2×C5
HallC72×C11 ⋊ C2×C3×C5
HallC2×C72 ⋊ C3×C11⋊C5
SylowC72 ⋊ C2×C3×C11⋊C5
HallC2×C11 ⋊ C5×C7×ΓL(1,𝔽8)
SylowC11 ⋊ C2×C5×C7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2×C7ΓL(1,𝔽8)×C11⋊C5
commutator, cosocle, hypocenterC7×C11C2×C3×C5×C7
FrattiniC1G
Fitting, socleC2×C72×C11C3×C5
LCSG ▷ C7×C11
UCSC1 ◁ C2×C7

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 462

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC7×ΓL(1,𝔽8)⋊(Dic11⋊C5), C7×C11⋊C5×ΓL(1,𝔽8)⋊C4, C7×ΓL(1,𝔽8)×Dic11⋊C5, …
maximal normal subgroup ofC7×ΓL(1,𝔽8)⋊(Dic11⋊C5), C7×C11⋊C5×ΓL(1,𝔽8)⋊C4, C7×ΓL(1,𝔽8)×Dic11⋊C5, …
maximal quotient ofC4×C7×ΓL(1,𝔽8)×C11⋊C5, C22×C7×ΓL(1,𝔽8)×C11⋊C5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 57);