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Q₈×D₂₁ #66731

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

Aliases

Q₈×D₂₁

Splittings

Fitting	C₃×C₇×Q₈ ⋊ C₂
Hall	C₃×C₇ ⋊ C₂×Q₈
Sylow	C₇ ⋊ D₃×Q₈
Sylow	C₃ ⋊ Q₈×D₇

Subgroups

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

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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10621067, #10621073, #10621084, …
maximal normal subgroup of	#10621067, #10621073, #10621084, …
maximal quotient of	#10621370, #10621371, #10621372, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 200);