C₂×D₁₆₁ #10619473

order	644
isoclinism	isoclinic class of D₁₆₁
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₂₂
direct product	C₂×D₁₆₁
unambiguous semidirect product	(C₂×C₇)⋊D₂₃
unambiguous semidirect product	(C₂×C₂₃)⋊D₇

Splittings

Fitting, socle	C₂×C₇×C₂₃ ⋊ C₂
derived, Hall	C₇×C₂₃ ⋊ C₂²
Sylow	C₂₃ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₆₁
commutator, cosocle, hypocenter	C₇×C₂₃	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₂₃	C₂
LCS	G ▷ C₇×C₂₃
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 322

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2)(3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(4,9)(5,8)(6,7)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₆₁⋊C₄, #13046227, #13046228, …
maximal normal subgroup of	D₁₆₁⋊C₄, #13046227, #13046228, …
maximal quotient of	Dic₃₂₂, C₄×D₁₆₁, D₆₄₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(644, 8);

C2×D161 #10619473

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×D₁₆₁ #10619473