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D₇×Dic₆ #66668

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

Aliases

direct product	D₇×Dic₆
unambiguous semidirect product	(C₃×D₇)⋊Q₈

Splittings

Hall	C₃×C₇ ⋊ C₂×Q₈
Sylow	C₇ ⋊ C₂×Dic₆
Sylow	C₃ ⋊ Q₈×D₇

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×C₃×D₇	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:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#204, D₇×Dic₁₂, #10620822, …
maximal normal subgroup of	D₇×#204, D₇×Dic₁₂, #10620822, …
maximal quotient of	#10620885, #10620890, #10620891, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 137);