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D₃×Dic₁₄ #66671

order	336
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

direct product	D₃×Dic₁₄
unambiguous semidirect product	(D₃×C₇)⋊Q₈

Splittings

Hall	C₃×C₇ ⋊ C₂×Q₈
Sylow	C₇ ⋊ D₃×Q₈
Sylow	C₃ ⋊ C₂×Dic₁₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₇
commutator, socle, cosocle	C₂×C₃×C₇	C₂³
Fitting	C₃×C₄×C₇	C₂²
prime radical	C₂×D₃×C₇	C₂²
hypercenter	C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2,4,6,7,5,3)(8,10,11,15)(9,12,13,14)(16,17,18);(2,3)(4,5)(6,7)(8,9,11,13)(10,14,15,12);(17,18)

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1132, D₃×Dic₂₈, #10620825, …
maximal normal subgroup of	D₃×#1132, D₃×Dic₂₈, #10620825, …
maximal quotient of	#10620886, #10620889, #10620890, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 140);