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D₃×C₁₃×C₇⋊D₇ #21439765

order	7644
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₃ · Schur multiplier (H₂) = C₂×C₇

Aliases

direct product	D₃×C₁₃×C₇⋊D₇
unambiguous semidirect product	(D₃×C₇)⋊(C₁₃×D₇)

Splittings

Fitting, socle, Hall	C₃×C₇²×C₁₃ ⋊ C₂²
derived, Hall	C₃×C₇² ⋊ C₂²×C₁₃
Hall	C₇²×C₁₃ ⋊ C₂×D₃
Sylow	C₇² ⋊ C₂×D₃×C₁₃
Hall	C₃×C₁₃ ⋊ C₂×C₇⋊D₇
Sylow	C₃ ⋊ C₂×C₁₃×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃	D₃×C₇⋊D₇
commutator, hypocenter	C₃×C₇²	C₂²×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₃	C₂²
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₁₃

200 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1014 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₃×D₉×C₇⋊D₇, D₃×C₁₃×ΓL(1,𝔽₈)⋊D₇, D₃×#16838471, …
maximal normal subgroup of	D₃×C₁₃×ΓL(1,𝔽₈)⋊D₇, D₃×#16838471, D₃×#16838470, …
maximal quotient of	C₁₃×D₉×C₇⋊D₇, C₃×D₃×C₁₃×C₇⋊D₇, D₃×C₁₃×C₃⋊(C₇⋊D₇), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7644, 334);