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C₂×#4248 #66691

order	336
isoclinism	isoclinic class of #4245
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

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

Subgroups

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

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:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620885, #10620886, #10620887, …
maximal normal subgroup of	#10620885, #10620886, #10620887, …
maximal quotient of	#10620943, #10620946, #10620950, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 160);