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D₂₁⋊D₆₇ #19150620

order	5628
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₂₁⋊D₆₇
unambiguous semidirect product	D₂₀₁⋊D₇
unambiguous semidirect product	D₄₆₉⋊D₃

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₆₇ ⋊ C₂²
Hall	C₇×C₆₇ ⋊ C₂×D₃
Hall	C₃×C₆₇ ⋊ C₂×D₇
Sylow	C₆₇ ⋊ D₃×D₇
Hall	C₃×C₇ ⋊ C₂×D₆₇
Sylow	C₇ ⋊ D₃×D₆₇
Sylow	C₃ ⋊ D₇×D₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₇×C₆₇	C₂²
LCS	G ▷ C₃×C₇×C₆₇
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,7,67|d^a=d^6,e^a=e^66,c^b=c^2,d^b=d^6

Conjugacy classes & character table

411 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₆₃⋊D₆₇, #14584204⋊D₃, #14584203⋊D₃, …
maximal normal subgroup of	#14584204⋊D₃, #14584203⋊D₃, (C₃⋊AGL(1,𝔽₇))⋊D₆₇, …
maximal quotient of	D₆₃⋊D₆₇, C₃×D₂₁⋊D₆₇, D₄₆₉⋊(C₃⋊D₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5628, 48);