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C₂×#1204 #6787

order	240
isoclinism	isoclinic class of #1201
automorphism	Aut(G) = D₃×AGL(1,𝔽₅)×#120 (order 3840) · Inn(G) = D₃×D₅ · Out(G) = #525
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Hall	C₃×C₅ ⋊ C₂×Q₈
Sylow	C₅ ⋊ C₂×Dic₆
Sylow	C₃ ⋊ C₂×Dic₁₀

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₃×D₅
commutator, cosocle	C₂×C₃×C₅	C₂³
Frattini	C₂	C₂×D₃×D₅
Fitting, socle, prime radical	C₂²×C₃×C₅	C₂²
hypocenter	C₃×C₅	C₂×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#91175, #91323, #91324, …
maximal normal subgroup of	#91175, #91323, #91324, …
maximal quotient of	#91381, #91384, #91388, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 148);