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D₁₁₉⋊D₂₁ #23756258

order	9996
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

D₁₁₉⋊D₂₁

Splittings

Fitting, socle, derived, Hall	C₃×C₇²×C₁₇ ⋊ C₂²
Hall	C₇²×C₁₇ ⋊ C₂×D₃
Hall	C₃×C₇² ⋊ C₂×D₁₇
Hall	C₃×C₁₇ ⋊ D₇²
Sylow	C₇² ⋊ D₃×D₁₇
Sylow	C₁₇ ⋊ D₇×D₂₁
Sylow	C₃ ⋊ (C₇⋊D₇)⋊D₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇²×C₁₇	C₂²
LCS	G ▷ C₃×C₇²×C₁₇
UCS	C₁

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,7^2,17|d^a=d^6,f^a=f^16,c^b=c^2,d^b=d^6,e^b=e^6

Conjugacy classes & character table

660 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₁₉⋊D₆₃, #44472781, #44472802, …
maximal normal subgroup of	#44472802, (C₁₇⋊AGL(1,𝔽₇))⋊D₂₁, #44472833, …
maximal quotient of	D₁₁₉⋊D₆₃, C₃×D₁₁₉⋊D₂₁, #44472966, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 161);