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C112×C22⋊ΓL(1,𝔽8) #23779072

order10164
isoclinismisoclinic class of C22⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C112 · Schur multiplier (H2) = C2×C11

Aliases

direct productC112×C22⋊ΓL(1,𝔽8)
unambiguous semidirect productC22⋊(C112×ΓL(1,𝔽8))
unambiguous semidirect productC7⋊(C112×A4)
unambiguous semidirect product(C22×C11)⋊(C11×ΓL(1,𝔽8))
unambiguous semidirect product(C7×C11)⋊(C11×A4)

Splittings

Fitting, socle, HallC22×C7×C112 ⋊ C3
derived, HallC22×C7 ⋊ C3×C112
HallC7×C112 ⋊ A4
HallC22×C112 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ C112×A4
SylowC22 ⋊ C112×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC112C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C3×C112
FrattiniC1G
Fitting, socleC22×C7×C112C3
LCSG ▷ C22×C7
UCSC1 ◁ C112

140 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1452 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC22⋊ΓL(1,𝔽8)×C112⋊C3, (C22⋊ΓL(1,𝔽8))⋊(C112⋊C3), C112×A4×ΓL(1,𝔽8), …
maximal normal subgroup ofC22⋊ΓL(1,𝔽8)×C112⋊C3, (C22⋊ΓL(1,𝔽8))⋊(C112⋊C3), C112×A4×ΓL(1,𝔽8), …
maximal quotient ofC112×C22⋊(C7⋊C9), C3×C112×C22⋊ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10164, 124);