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C7×C22⋊(C112⋊C3) #23779073

order10164
automorphism?
homologyAbelianization (H1) = C3×C7 · Schur multiplier (H2) = C2×C11

Aliases

direct productC7×C22⋊(C112⋊C3)
unambiguous semidirect productC22⋊(C7×C112⋊C3)
unambiguous semidirect productC112⋊(C7×A4)

Splittings

Fitting, socle, HallC22×C7×C112 ⋊ C3
derived, HallC22×C112 ⋊ C3×C7
HallC7×C112 ⋊ A4
SylowC112 ⋊ C7×A4
HallC22×C7 ⋊ C112⋊C3
SylowC22 ⋊ C7×C112⋊C3

Subgroups

subgroupG/H
center, hypercenterC7C22⋊(C112⋊C3)
commutator, cosocle, hypocenterC22×C112C3×C7
FrattiniC1G
Fitting, socleC22×C7×C112C3
LCSG ▷ C22×C112
UCSC1 ◁ C7

60 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1148 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×A4×C112⋊C3, ΓL(1,𝔽8)×C22⋊(C112⋊C3), (C22⋊ΓL(1,𝔽8))⋊(C112⋊C3), …
maximal normal subgroup ofC7×A4×C112⋊C3, ΓL(1,𝔽8)×C22⋊(C112⋊C3), (C22⋊ΓL(1,𝔽8))⋊(C112⋊C3), …
maximal quotient ofC7×C22⋊(C112⋊C9), C3×C7×C22⋊(C112⋊C3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10164, 125);