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C₂×#4268 #66734

order	336
isoclinism	isoclinic class of Dic₄₂
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₂³×C₃×C₇ ⋊ C₂
Hall	C₃×C₇ ⋊ C₂×D₄
Sylow	C₇ ⋊ C₂×#67
Sylow	C₃ ⋊ C₂×C₂²⋊D₇

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₁
commutator, cosocle	C₂×C₃×C₇	C₂³
Frattini	C₂	C₂²×D₂₁
Fitting	C₂³×C₃×C₇	C₂
socle	C₂²×C₃×C₇	C₂²
hypercenter	C₂³	D₂₁
hypocenter	C₃×C₇	C₂×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

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:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620532, #10620645, #10621088, …
maximal normal subgroup of	#10620532, #10620645, #10621088, …
maximal quotient of	C₂×#66629, #10621407, C₂×#66631, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 203);