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C7×(C11⋊Dic11)⋊C3 #23778985

order10164
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C11

Aliases

direct productC7×(C11⋊Dic11)⋊C3
unambiguous semidirect product(C11⋊Dic11)⋊(C3×C7)

Splittings

derived, SylowC112 ⋊ C3×C4×C7
HallC7×C11⋊Dic11 ⋊ C3
HallC7×C112⋊C3 ⋊ C4
HallC7×C112 ⋊ C3×C4
HallC11⋊Dic11 ⋊ C3×C7
HallC112⋊C3 ⋊ C4×C7

Subgroups

subgroupG/H
center, hypercenterC2×C7(C11⋊D11)⋊C3
commutator, hypocenterC112C3×C4×C7
FrattiniC2C7×(C11⋊D11)⋊C3
Fitting, socleC2×C7×C112C2×C3
prime radicalC2×C7×C112⋊C3C2
cosocleC2×C112C2×C3×C7
LCSG ▷ C112
UCSC1 ◁ C2×C7

48 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×(C11⋊Dic11)⋊C3, C3×C7×(C11⋊Dic11)⋊C3, C7×C3⋊((C11⋊Dic11)⋊C3)
maximal normal subgroup ofΓL(1,𝔽8)×(C11⋊Dic11)⋊C3, C3×C7×(C11⋊Dic11)⋊C3
maximal quotient ofC7×(C11⋊Dic11)⋊C9, C3×C7×(C11⋊Dic11)⋊C3, C7×C3⋊((C11⋊Dic11)⋊C3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10164, 37);