Finite Group Databasebeta

← Prev (#66689) Next (#66691) →

C₂×#4247 #66690

order	336
isoclinism	isoclinic class of #4245
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Hall	C₃×C₇ ⋊ C₂×D₄
Sylow	C₇ ⋊ C₂×D₁₂
Sylow	C₃ ⋊ C₂×C₂²⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₃×D₇
commutator, cosocle	C₂×C₃×C₇	C₂³
Frattini	C₂	C₂×D₃×D₇
Fitting, socle	C₂²×C₃×C₇	C₂²
hypocenter	C₃×C₇	C₂×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂²

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2)(4,5)(7,9)(10,11,13,15,16,14,12);(3,4,5)(6,7,8,9)(11,12)(13,14)(15,16);(1,2)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620888, #10620912, #10620913, …
maximal normal subgroup of	#10620888, #10620912, #10620913, …
maximal quotient of	C₂×#66562, #10620865, #10620866, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 159);