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C₃×C₂²⋊AGL(1,𝔽₅) #6756

order	240
isoclinism	isoclinic class of #700
automorphism	Aut(G) = C₂×D₄×AGL(1,𝔽₅) (order 320) · Inn(G) = C₂×AGL(1,𝔽₅) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃×C₂²⋊AGL(1,𝔽₅)
unambiguous semidirect product	(C₂×AGL(1,𝔽₅))⋊(C₂×C₃)

Splittings

Fitting	C₂²×C₃×C₅ ⋊ C₄
Hall	C₃×C₅ ⋊ C₂²⋊C₄
Sylow	C₅ ⋊ C₃×C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄
Frattini	C₂	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₂²×C₃×C₅	C₄
socle	C₂×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂
cosocle	C₂×D₅	C₂²×C₃
hypercenter	C₂²×C₃	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#91172, #91173, C₃×#3999, …
maximal normal subgroup of	#91172, #91173, C₃×#3999, …
maximal quotient of	C₃×#3999, C₃×#4003, C₃×#4005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 117);