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C₂×Dic₄₂ #66725

order	336
isoclinism	isoclinic class of Dic₄₂
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

C₂×Dic₄₂

Splittings

Hall	C₃×C₇ ⋊ C₂×Q₈
Sylow	C₇ ⋊ C₂×Dic₆
Sylow	C₃ ⋊ C₂×Dic₁₄

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₁
commutator, cosocle	C₂×C₃×C₇	C₂³
Frattini	C₂	C₂²×D₂₁
Fitting	C₂×C₃×C₄×C₇	C₂
socle, prime radical	C₂²×C₃×C₇	C₂²
hypercenter	C₂×C₄	D₂₁
hypocenter	C₃×C₇	C₂×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 20 points:

(3,4,5)(6,7,9,11,12,10,8)(13,15,16,20)(14,17,18,19);(1,2)(4,5)(7,8)(9,10)(11,12)(13,14,16,18)(15,19,20,17);(1,2)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620498, #10620509, #10620510, …
maximal normal subgroup of	#10620498, #10620509, #10620510, …
maximal quotient of	C₄×Dic₄₂, #10621352, #10621353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 194);