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D₅⋊(C₃⋊C₈) #6757

order	240
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	Aut(G) = D₃×D₄×AGL(1,𝔽₅) (order 960) · Inn(G) = ΓL(1,𝔽₁₆) · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

D₅⋊(C₃⋊C₈)

Splittings

derived, Hall	C₃×C₅ ⋊ C₂×C₈
Sylow	C₅ ⋊ C₂×C₃⋊C₈
Sylow	C₃ ⋊ D₅⋊C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂×C₈
Frattini	C₂	C₂×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄×C₅	C₄
socle	C₂×C₃×C₅	C₂×C₄
prime radical	C₂×C₃×D₅	C₄
cosocle	C₃×Dic₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₄

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2^4,3,5|a^2=c,b^2=d,c^2=d,e^a=e^2,f^a=f^3,f^c=f^4

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₃⋊C₈, #91149, #91151, …
maximal normal subgroup of	AGL(1,𝔽₅)×C₃⋊C₈, #91149, #91151, …
maximal quotient of	D₅⋊(C₃⋊C₁₆), #91219, C₄×C₃⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 118);