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Q₈×D₁₅ #6820

order	240
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₅)×S₄ (order 5760) · Inn(G) = C₂²×D₁₅ · Out(G) = C₂×C₄×D₃
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×D₁₅

Splittings

Fitting	C₃×C₅×Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ C₂×Q₈
Sylow	C₅ ⋊ D₃×Q₈
Sylow	C₃ ⋊ Q₈×D₅

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₃×C₅×Q₈	C₂
prime radical	C₂×D₁₅	C₂²
hypercenter	Q₈	D₁₅
hypocenter	C₃×C₅	C₂×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ Q₈

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#91505, #91511, #91522, …
maximal normal subgroup of	#91505, #91511, #91522, …
maximal quotient of	#91776, #91777, #91778, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 181);