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C₅×#204 #6690

order	240
isoclinism	isoclinic class of #204
automorphism	Aut(G) = C₂×C₄×D₃×D₄ (order 384) · Inn(G) = D₁₂ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

Aliases

general semilinear	ΓL(1,𝔽₁₂₁)
direct product	C₅×#204

Splittings

Fitting	C₃×C₅×C₈ ⋊ C₂
Hall	C₃×C₅ ⋊ SD₁₆
Sylow	C₃ ⋊ C₅×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₅	D₁₂
commutator, cosocle	C₃×C₄	C₂²×C₅
Frattini	C₄	C₂×C₅×D₃
Fitting	C₃×C₅×C₈	C₂
socle	C₂×C₃×C₅	D₄
prime radical	C₅×D₁₂	C₂
hypercenter	C₅×C₈	D₃
hypocenter	C₃	C₅×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈

40 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:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#204, #91249, #91255, …
maximal normal subgroup of	D₅×#204, #91249, #91255, …
maximal quotient of	C₅×#816, C₅×#817, C₅×#821, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 51);