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C₅×#239 #6811

order	240
automorphism	Aut(G) = C₂×C₄×D₃×S₄ (order 1152) · Inn(G) = C₂²×D₃ · Out(G) = C₂×C₄×D₃
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₃×C₅×Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #41
Sylow	C₃ ⋊ C₅×#41

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂³×C₅
Frattini	C₂	C₂²×C₅×D₃
Fitting	C₃×C₅×Q₈	C₂
socle	C₂×C₃×C₅	C₂³
hypercenter	C₅×Q₈	D₃
hypocenter	C₃	C₅×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₅×Q₈

80 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:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#91504, #91510, #91517, …
maximal normal subgroup of	#91504, #91510, #91517, …
maximal quotient of	C₅×#890, C₅×#892, C₅×#893, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 172);