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Q₈×ΓL(1,𝔽₁₆) #91993

order	480
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×ΓL(1,𝔽₁₆)

Splittings

Fitting	C₃×C₅×Q₈ ⋊ C₄
Hall	C₃×C₅ ⋊ C₄×Q₈
Sylow	C₅ ⋊ Q₈×Dic₃
Sylow	C₃ ⋊ Q₈×AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅	C₂²×C₄
Fitting	C₃×C₅×Q₈	C₄
prime radical, cosocle	C₂×C₃×D₅	C₂³
hypercenter	Q₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₄×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ Q₈

140 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);(1,2,5,3,4)(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	#11751965, #11751966, #11751969, …
maximal normal subgroup of	#11751965, #11751966, #11751969, …
maximal quotient of	#11752362, #11752366, #11752368, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1069);