Finite Group Databasebeta

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D₁₆₈ #66624

order	336
isoclinism	isoclinic class of #66623
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

D₁₆₈

Splittings

Fitting	C₃×C₇×C₈ ⋊ C₂
Hall	C₃×C₇ ⋊ D₈
Sylow	C₇ ⋊ D₂₄
Sylow	C₃ ⋊ D₅₆

Subgroups

	subgroup	G/H
center	C₂	D₈₄
commutator, cosocle	C₃×C₄×C₇	C₂²
Frattini	C₄	C₂×D₂₁
Fitting	C₃×C₇×C₈	C₂
socle	C₂×C₃×C₇	D₄
hypercenter	C₈	D₂₁
hypocenter	C₃×C₇	D₈
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

2^4,3,7|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^6

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620473, #10620474, #10620478, …
maximal normal subgroup of	#10620473, #10620474, #10620478, …
maximal quotient of	D₃₃₆, #10620639, Dic₁₆₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 93);