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C₂×C₃×C₄×AGL(1,𝔽₅) #91974

order	480
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product	C₂×C₃×C₄×AGL(1,𝔽₅)
unambiguous semidirect product	D₅⋊(C₃×C₄²)

Splittings

Fitting	C₂×C₃×C₄×C₅ ⋊ C₄
derived, Sylow	C₅ ⋊ C₂×C₃×C₄²
Hall	C₃×C₅ ⋊ C₂×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₃×C₄²
Frattini	C₂	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₄×C₅	C₄
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂²
cosocle	C₂×D₅	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×AGL(1,𝔽₅))⋊(C₃⋊C₈), #11746526, #11746527, …
maximal normal subgroup of	(C₂×AGL(1,𝔽₅))⋊(C₃⋊C₈), #11746526, #11746527, …
maximal quotient of	C₃×C₄×D₅⋊C₈, C₃×#65676, C₃×C₄×#700, …
automorphism group of	C₁₃×Dic₅, C₂×D₅×C₁₃, C₅×C₇×D₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1050);