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C₁₁²×C₇⋊C₉ #21439311

order	7623
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₁² · Schur multiplier (H₂) = C₁₁

Aliases

direct product	C₁₁²×C₇⋊C₉
unambiguous semidirect product	C₇⋊(C₉×C₁₁²)
unambiguous semidirect product	(C₇×C₁₁)⋊(C₉×C₁₁)

Splittings

derived, Sylow	C₇ ⋊ C₉×C₁₁²
Hall	C₇×C₁₁² ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₁²	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₉×C₁₁²
Frattini	C₃	C₁₁²×ΓL(1,𝔽₈)
Fitting, socle	C₃×C₇×C₁₁²	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₁₁²

84 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1815 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₁²×D₇⋊C₉, C₁₁×C₁₁⋊(D₇⋊C₉), C₁₁⋊(C₁₁⋊(D₇⋊C₉)), …
maximal normal subgroup of	C₁₁²×D₇⋊C₉, C₁₁×C₁₁⋊(D₇⋊C₉), C₁₁⋊(C₁₁⋊(D₇⋊C₉)), …
maximal quotient of	C₂×C₁₁²×C₇⋊C₉, C₁₁²×C₂²⋊(C₇⋊C₉), C₅×C₁₁²×C₇⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7623, 7);