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D₇×#67 #66692

order	336
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

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

Subgroups

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

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#235, #10621564, D₃×D₄×D₇, …
maximal normal subgroup of	D₇×#235, #10621564, D₃×D₄×D₇, …
maximal quotient of	#10620888, #10620889, #10620910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 161);