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D₄×D₂₁ #66729

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

Aliases

D₄×D₂₁

Splittings

Fitting	C₃×C₇×D₄ ⋊ C₂
Hall	C₃×C₇ ⋊ C₂×D₄
Sylow	C₇ ⋊ D₃×D₄
Sylow	C₃ ⋊ D₄×D₇

Subgroups

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

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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10621043, #10621049, #10621060, …
maximal normal subgroup of	#10621043, #10621049, #10621060, …
maximal quotient of	#10621361, C₂²⋊C₄×D₂₁, #10621364, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 198);