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C₇×D₇×D₁₃⋊C₃ #21439596

order	7644
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×D₇×D₁₃⋊C₃
unambiguous semidirect product	D₁₃⋊(C₃×C₇×D₇)
unambiguous semidirect product	(C₇×D₁₃⋊C₃)⋊(C₂×C₇)
unambiguous semidirect product	(D₇×C₁₃⋊C₃)⋊(C₂×C₇)
unambiguous semidirect product	((C₁₃⋊C₃)⋊D₇)⋊(C₂×C₇)

Splittings

Fitting, socle, Hall	C₇²×C₁₃ ⋊ C₂²×C₃
derived	C₇×C₁₃ ⋊ C₂²×C₃×C₇
Hall	C₇×D₇×D₁₃ ⋊ C₃
Hall	C₇²×C₁₃⋊C₃ ⋊ C₂²
Sylow	C₇² ⋊ C₂×D₁₃⋊C₃
Hall	C₁₃⋊C₃ ⋊ C₂×C₇×D₇
Sylow	C₁₃ ⋊ C₂×C₃×C₇×D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₇×D₁₃⋊C₃
commutator, hypocenter	C₇×C₁₃	C₂²×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₁₃	C₂²×C₃
LCS	G ▷ C₇×C₁₃
UCS	C₁ ◁ C₇

104 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₃⋊C₃×#64418, C₇×AGL(1,𝔽₇)×D₁₃⋊C₃, D₇×ΓL(1,𝔽₈)×D₁₃⋊C₃, …
maximal normal subgroup of	C₇×AGL(1,𝔽₇)×D₁₃⋊C₃, D₇×ΓL(1,𝔽₈)×D₁₃⋊C₃, D₁₃⋊C₃×#64423, …
maximal quotient of	C₇×D₇×D₁₃⋊C₉, C₃×C₇×D₇×D₁₃⋊C₃, C₇×D₂₁×D₁₃⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7644, 165);